There are tears in my jaded eyes
As I walk through the woods of dark greens
With soft breezes speaking to the trees
I can see the falling buildings in my heart
As blotches of undoings around me
The cold, dark night unfolds its mystery
As I saunter through its mind
I can feel ulterior lasers pierce me
My reflection sliding through their eyes
With every footstep, there goes my insulation
When I hear you, I can feel all my pain
When I hear you, I can feel all my shame
In the distance, over the soft mound
I hear the music of all things past
Sweet melodies, wind and waters
Guiding me in and out of those memories
Of twisting and obscure stagnations
Of speckled tears
When I hear you, I can feel all my downfalls
When I hear you, I can feel all my advances
My eyes catch sight of the deep blue sky
Violet wavelets meeting melancholy oranges
Gentle pinks hold onto lurid purples
The sun leaves its memory indented in the sky
I hear the sounds of light touching the clouds
And I hear her whisper to me
When I hear you, I can feel all my ugliness
When I hear you, I can feel all my tenderness
So sweet and tender, like mother and child
The colours hold on and play
Round and round, the whites and violets tumble
Up and down, they dance to the sun's memory
She whispers with the dancing colours
When I hear you, I can feel all my feelings
When I hear you, I can feel all my life
I can feel myself flying in the water
As every wave sings to the memory
I can feel the pain slowly subsiding
No bridges
No walls
No impurity
As my heart sings inside me
I begin to fall into the sky
Into the deep blue sky
Gliding and spinning through the clouds
I see the ocean of colours
As they make love to obscurity
As soft as the child's skin
I hear her whisper
When I hear you, I can feel all my agony
Disappear
There is no more evasion
No more late nights with tears in my pillow
No more lunchtime beatings
Together we strangle loneliness
No more nails
No more screaming
No more bleeding
No more isolation
No more, no more
No more again
No more, no more
No more again
When I hear you
When I hear you, I can feel all my reasons
When I hear you, existence has a reason
Watery orchestras and dances in the sky
My heart is covered by the blanket of love
When I hear you
Love is whispering to the world
Shhh, listen
As I levitate in the phantasmagoria of courting colours
Am I living to die, or am I dying to live?
Byron P. Brassel
Middlemist, IX
25 September 2012
Tuesday, September 25, 2012
Thursday, June 28, 2012
Moonlight Dance
Her hair moves and sways like a river as the wind travels through it, so
beautiful in its composition... its form... its entity. I am standing
alone, on a beach, listening to the melody of the dark, embrionic waves
caressing the soft and sensuous sand... listening to her feet as she
walks toward me. The moon lies, staring... composing the music that the
waves are performing. It is an eloquent harmony that travels through my
ears and massages me inside... I hear a whisper...
I watch her as she walks, and I feel her grace as she leaves only slight footprints in the sand... she is an angel, coming toward me, coming to me. It seems like an eternity before she reaches me but she does... I hear a whisper... She stops only a few centimetres away, where she can allow me the deference of looking into her eyes. Inside there I see the whispers, I hear her dreams, I feel her disposition and I listen, I listen to her life, her past, her present and her future, and I look for me... but I cannot see... I only hear a whisper.
I hold out my hand to her invitingly, beckoning her to take it, to hold it... my eyes beg as they look deeply into her ocean of dreams. My hand shakes as it senses hers arrive. She slowly runs her forefinger over my palm and makes a loop, I feel her compassion, courage, fear and love dissipating into me... I hear a whisper. She closes her hand around mine and holds it, tightly, never taking her eyes off mine. In her eyes I see infinity... and in her eyes infinity has an end.
She begins to walk me down the beach holding my hand close, her hair still flowing like a wild river. I lean my face close enough for her hair to flow into it and I feel it... I smell it... engulfing me, intoxicating me... what an aroma. She suddenly stops and turns to face me. Our eyes once again lock in an unbreakable embrace while her alluring blue and white dress hugs me and brings me closer to her. We are both trapped... I hear a whisper.
I bring my hand to her face and run the back of my finger down the side, silk has no comparison. Her eyes close to the sensation and she leans her face toward my hand, resting it there. Again I hear a whisper... where? We both listen to the harmony of the waves and she starts to move slowly, from side to side, still holding my hand, still resting on my hand. I hear the whisper becoming louder, then silence... silence. I notice she is looking right into my eyes again, searchingly, looking for me. I take her other hand... gently, and I bring it to my face, breathing it in and then kissing it. Her eyes close to the sensation as the pleasure trickles its way into her, searching for every crevice and space, this pleasure expresses itself on her face as a sense of yearning travels across it. It is this expression of yearning, that I begin to fall in love with. Again I hear a whisper... but where?
I bring her closer to me, while her dress hugs me and brings me closer to her. I begin to move with her, in harmony with her as the moon conducts the ocean orchestra playing so beautifully for us. We are at one with nature as nature is at one with us, a complete unity of our souls. The moon carefully oversees the waves as they move, interfere and overlap, conducting crescendos and diminuendos as we waltz to the amiable sound that results from such a pure composition of nature. The water reaches our feet and wanders in-between, searching, covering and soothing. Our feet glide across the water as it recapitulates its advancement and recession, slowly and quickly with us as we do the same with our graceful movements. Suddenly, all grace is lost as we both break into laughter for one of my mistakes in stepping on her feet. It’s a beautiful moment. We stand for a moment and glance up at the moon, who seems disappointed that we have stopped our dance to his music. He looks back at us with the dark purple and pink clouds lighting up with him, almost clothing him, whilst still conducting his faithful 4 billion year old orchestra in the longest standing soundtrack in history. A special understanding develops between us and the moon as we stand and gaze at it, a great respect. My beautiful partner then rests her head on me and puts her arms around me, holding on, not wanting this moment to end... ever. We slowly move to the music again and... I hear a whisper.
I move with her and breathe in the pleasant aroma of her hair and my grip around her tightens slightly as my love for her increases. Am I in a dream, I ask myself quietly in my head. How could I have been allowed such an honour, of being allowed to guide so beautiful a woman? I try not to think in this gorgeous moment and I suddenly feel her grip tighten and then release, and she looks deeply into my abyss of dreams, and I see, I finally see. I am the whisperer...
A tear slowly trickles down my face, leaving a permanent mark engraved there as she continues to look in my eyes, sympathetic, enlightened and loving. I bring her face close to mine, where we are breathing in each other, tasting each other’s breath. She closes her lips over my mark as my eyes close to the sweetest sensation of the warmth and softness of her delicate kiss. We then look at each other’s lips and she bites her bottom lip, almost in practice for what is to come. I am in desperation, waiting, dying, for it to happen. She moves closer then pauses... there is silence, almost as if the waves stopped!
I then realise that I have to move too and I do. The embrace is powerful, awe-inspiring, warm, cold, soft, hard, magnetic, impulsive, impudent... amazing. I taste her as she tastes me and I let out more tears to the feeling of completeness. A sense and feeling I have never experienced, awakens in me like a fire, and for the first time, ever, I feel wanted.
She looks again into my eyes, finally, as the kiss ends, and rests her head on me again as we begin to sway again under the night sun... in our Moonlight Dance.
I whisper into her ears now saying: ‘May I be so lucky as to walk into your dreams and dance with you on this moonlit beach, and may I be so lucky that you walk into mine and dance with me’
Byron P. Brassel
23 September 2009
Re-mastered in the years since.
I watch her as she walks, and I feel her grace as she leaves only slight footprints in the sand... she is an angel, coming toward me, coming to me. It seems like an eternity before she reaches me but she does... I hear a whisper... She stops only a few centimetres away, where she can allow me the deference of looking into her eyes. Inside there I see the whispers, I hear her dreams, I feel her disposition and I listen, I listen to her life, her past, her present and her future, and I look for me... but I cannot see... I only hear a whisper.
I hold out my hand to her invitingly, beckoning her to take it, to hold it... my eyes beg as they look deeply into her ocean of dreams. My hand shakes as it senses hers arrive. She slowly runs her forefinger over my palm and makes a loop, I feel her compassion, courage, fear and love dissipating into me... I hear a whisper. She closes her hand around mine and holds it, tightly, never taking her eyes off mine. In her eyes I see infinity... and in her eyes infinity has an end.
She begins to walk me down the beach holding my hand close, her hair still flowing like a wild river. I lean my face close enough for her hair to flow into it and I feel it... I smell it... engulfing me, intoxicating me... what an aroma. She suddenly stops and turns to face me. Our eyes once again lock in an unbreakable embrace while her alluring blue and white dress hugs me and brings me closer to her. We are both trapped... I hear a whisper.
I bring my hand to her face and run the back of my finger down the side, silk has no comparison. Her eyes close to the sensation and she leans her face toward my hand, resting it there. Again I hear a whisper... where? We both listen to the harmony of the waves and she starts to move slowly, from side to side, still holding my hand, still resting on my hand. I hear the whisper becoming louder, then silence... silence. I notice she is looking right into my eyes again, searchingly, looking for me. I take her other hand... gently, and I bring it to my face, breathing it in and then kissing it. Her eyes close to the sensation as the pleasure trickles its way into her, searching for every crevice and space, this pleasure expresses itself on her face as a sense of yearning travels across it. It is this expression of yearning, that I begin to fall in love with. Again I hear a whisper... but where?
I bring her closer to me, while her dress hugs me and brings me closer to her. I begin to move with her, in harmony with her as the moon conducts the ocean orchestra playing so beautifully for us. We are at one with nature as nature is at one with us, a complete unity of our souls. The moon carefully oversees the waves as they move, interfere and overlap, conducting crescendos and diminuendos as we waltz to the amiable sound that results from such a pure composition of nature. The water reaches our feet and wanders in-between, searching, covering and soothing. Our feet glide across the water as it recapitulates its advancement and recession, slowly and quickly with us as we do the same with our graceful movements. Suddenly, all grace is lost as we both break into laughter for one of my mistakes in stepping on her feet. It’s a beautiful moment. We stand for a moment and glance up at the moon, who seems disappointed that we have stopped our dance to his music. He looks back at us with the dark purple and pink clouds lighting up with him, almost clothing him, whilst still conducting his faithful 4 billion year old orchestra in the longest standing soundtrack in history. A special understanding develops between us and the moon as we stand and gaze at it, a great respect. My beautiful partner then rests her head on me and puts her arms around me, holding on, not wanting this moment to end... ever. We slowly move to the music again and... I hear a whisper.
I move with her and breathe in the pleasant aroma of her hair and my grip around her tightens slightly as my love for her increases. Am I in a dream, I ask myself quietly in my head. How could I have been allowed such an honour, of being allowed to guide so beautiful a woman? I try not to think in this gorgeous moment and I suddenly feel her grip tighten and then release, and she looks deeply into my abyss of dreams, and I see, I finally see. I am the whisperer...
A tear slowly trickles down my face, leaving a permanent mark engraved there as she continues to look in my eyes, sympathetic, enlightened and loving. I bring her face close to mine, where we are breathing in each other, tasting each other’s breath. She closes her lips over my mark as my eyes close to the sweetest sensation of the warmth and softness of her delicate kiss. We then look at each other’s lips and she bites her bottom lip, almost in practice for what is to come. I am in desperation, waiting, dying, for it to happen. She moves closer then pauses... there is silence, almost as if the waves stopped!
I then realise that I have to move too and I do. The embrace is powerful, awe-inspiring, warm, cold, soft, hard, magnetic, impulsive, impudent... amazing. I taste her as she tastes me and I let out more tears to the feeling of completeness. A sense and feeling I have never experienced, awakens in me like a fire, and for the first time, ever, I feel wanted.
She looks again into my eyes, finally, as the kiss ends, and rests her head on me again as we begin to sway again under the night sun... in our Moonlight Dance.
I whisper into her ears now saying: ‘May I be so lucky as to walk into your dreams and dance with you on this moonlit beach, and may I be so lucky that you walk into mine and dance with me’
Byron P. Brassel
23 September 2009
Re-mastered in the years since.
The Sound Of Her Skin
My head is in her soft flowing hair. A smell so natural, so
intoxicating... slowly creeps up my nose like a flowing river. She is in
my arms while we lie silently on the watery silk sheets. We are in our
purest states... naked to nature... naked to the world. A beautiful light
shines on her skin... making it glisten in a way that competes with a
star. The light travels and almost bounces off her perfect skin. She is
so beautiful... a beauty so soft and so tender, it could melt the heart
of the sun.
I run my finger slowly over her arm, listening to the sweet sound of her skin against mine, and the sound of her breath as she enjoys the sensation, the slow drive... the affectionate love. I can hear her smile... as my fingers continue to caress her gorgeous arm, feeling her dentless, velvety skin as it moves beneath me. Her back is against my stomach, and I feel her draw nearer as a light breeze engulfs us in its cool hands. I move my mouth over her shoulder, slowly and searchingly... breathing in all the time, what an aroma. I slowly close my lips over the edge of her shoulder and give her a soft and affectionate kiss, and I can feel her skin beneath me culminate as the pleasure she feels is over whelming... over powering...
She takes my arm and places it on her stomach and she plays with my fingers... she is in love with my hands as I am in love with everything she is. I move my hand over her stomach and she moves to the feeling. My finger creates a circle around her belly button and I can feel her exert a small giggle... my heart opens up and a smile travels across my face. The sound of our skin is reminiscent of the softest of waves falling on a beach. My hands move up over her stomach... over her shaded left breast... I feel her, hear her, smell her, see her... as she reacts in the softest and most loving way. She holds my hand there. I can feel her heart beating, as she can feel mine through her back... drums beating my life into her and her life into me. My head moves over her hair, searching for her ear. I whisper sweet secrets to her in the softest of tones where my voice becomes the wind that roams through her. I nibble on the outskirts of her ear and her giggle becomes a laugh.
She turns to face me. A face so exquisite looks into me. I see her in all her splendour. I look into her eyes and I see a galaxy... I see a world so large, a space... I see her dreams and into her heart... her life before, now and after... her eyes are an abyss of splendour and hold the keys to beauty’s definitive. She puts her arms around me and our faces are about a few millimetres apart. We are breathing each other, our eyes focused on each other... locked in an immortal love. Our noses touch, almost as if they too were in love with each other. I move my face over hers, playfully and teasingly... searching for her lips. I kiss her cheek and we smile... the search continues, slowly and lovingly and she is enjoying it more than ever. She searches too and we find what we are looking for. Our lips lock in an unbelievable embrace, I can feel the molecules of pleasure travel up my spine like a slow lightning and feel her grip around me tighten. My eyes close to an amplified passion and amplified love as our lips are at war with each other... the sensation travelling through us like a fire... a burning flame. Our breathing has become much faster now, still breathing each other. I can smell her breath and it in itself creates so much pleasure. Her whole life and my whole life and our lives are in this one sublime kiss. Almost as if this is the last kiss we will ever share... in eternity. I can feel the tears leaking their way down my face... a love so powerful... so amazing... a dream that is real... surreal... her lips: hard and soft... fluctuating in feeling... so amazingly ravishing.
The kiss ends as slowly and lovingly as it began and she looks into my watery eyes... and sheds a tear herself while wiping mine away. The tears have left a mark down my face. I hold my true love tightly and she returns the gesture, holding on for dear life. She closes her eyes and rests her face on mine and we lie there on and in each other. There is nothing else in the world but us... and the sound of her skin.
Byron P. Brassel
1 May 2009
I run my finger slowly over her arm, listening to the sweet sound of her skin against mine, and the sound of her breath as she enjoys the sensation, the slow drive... the affectionate love. I can hear her smile... as my fingers continue to caress her gorgeous arm, feeling her dentless, velvety skin as it moves beneath me. Her back is against my stomach, and I feel her draw nearer as a light breeze engulfs us in its cool hands. I move my mouth over her shoulder, slowly and searchingly... breathing in all the time, what an aroma. I slowly close my lips over the edge of her shoulder and give her a soft and affectionate kiss, and I can feel her skin beneath me culminate as the pleasure she feels is over whelming... over powering...
She takes my arm and places it on her stomach and she plays with my fingers... she is in love with my hands as I am in love with everything she is. I move my hand over her stomach and she moves to the feeling. My finger creates a circle around her belly button and I can feel her exert a small giggle... my heart opens up and a smile travels across my face. The sound of our skin is reminiscent of the softest of waves falling on a beach. My hands move up over her stomach... over her shaded left breast... I feel her, hear her, smell her, see her... as she reacts in the softest and most loving way. She holds my hand there. I can feel her heart beating, as she can feel mine through her back... drums beating my life into her and her life into me. My head moves over her hair, searching for her ear. I whisper sweet secrets to her in the softest of tones where my voice becomes the wind that roams through her. I nibble on the outskirts of her ear and her giggle becomes a laugh.
She turns to face me. A face so exquisite looks into me. I see her in all her splendour. I look into her eyes and I see a galaxy... I see a world so large, a space... I see her dreams and into her heart... her life before, now and after... her eyes are an abyss of splendour and hold the keys to beauty’s definitive. She puts her arms around me and our faces are about a few millimetres apart. We are breathing each other, our eyes focused on each other... locked in an immortal love. Our noses touch, almost as if they too were in love with each other. I move my face over hers, playfully and teasingly... searching for her lips. I kiss her cheek and we smile... the search continues, slowly and lovingly and she is enjoying it more than ever. She searches too and we find what we are looking for. Our lips lock in an unbelievable embrace, I can feel the molecules of pleasure travel up my spine like a slow lightning and feel her grip around me tighten. My eyes close to an amplified passion and amplified love as our lips are at war with each other... the sensation travelling through us like a fire... a burning flame. Our breathing has become much faster now, still breathing each other. I can smell her breath and it in itself creates so much pleasure. Her whole life and my whole life and our lives are in this one sublime kiss. Almost as if this is the last kiss we will ever share... in eternity. I can feel the tears leaking their way down my face... a love so powerful... so amazing... a dream that is real... surreal... her lips: hard and soft... fluctuating in feeling... so amazingly ravishing.
The kiss ends as slowly and lovingly as it began and she looks into my watery eyes... and sheds a tear herself while wiping mine away. The tears have left a mark down my face. I hold my true love tightly and she returns the gesture, holding on for dear life. She closes her eyes and rests her face on mine and we lie there on and in each other. There is nothing else in the world but us... and the sound of her skin.
Byron P. Brassel
1 May 2009
Saturday, June 23, 2012
Red River
He looked up. The cold
wind was blowing on his naked body. His hands, incarcerated by chains of
steel, were shaking, almost uncontrollably. He was cold, he was hurt,
he was tired, after a sleepless night and yet, his eyes focused on one
thing... his mother, the only human he saw. A look of complete serenity
and loyalty in his sore eyes, a look of complete and utter love, on the
verge of crying, washed across his face. His mother looked back at him,
knowing what was about to happen, anticipating the pain of his upcoming
suffering, yet the love in her eyes was unparalleled. She feared, yet
she comforted.
He could hear them, monsters holding tools of pain, preparing, reveling, to use the results of their wicked inventions. Yet he began praying for them, asking for forgiveness for their sins. The silent, powerful presence he evoked was ever so evident, yet he said nothing... he did not need to. His silence now, was his strength. The moment was drawing near; the soldiers were in position, waiting for the signal, the signal that would unleash the pain of the world. It came.
Unus... He could feel the pain slither up his spine, as the cane slapped his back... claw its way up his neck and in his heart, a ruthless, evil, searing pain. He made no cry, his face told the story, his eyes closed tight, clenched teeth, waiting... Duo... the pain doubled, his legs wet with a painful sweat, his back was arched, he couldn’t move and the cold breeze cut through the valley of his scars. Tres... he clenched his fists, the searing pain travelled through him, the journey to save the world, begun, how great is his love, that he would endure the torture, the pain, the suffering... how great is his love, that he would willingly walk into the shadow of all evil, to save a forsaken world. Blow after blow he received from the cane, the slapping noise echoed across the Preatorium, his eyes, wet with the tears, his hands shaking, his body shaking, his legs, barely able to hold him up. More blows were delivered like rain drops falling from the clouds... his legs gave way, his weak knees land on the cold stone floor, he could feel his open scars screaming the pain of the world, a forsaken pain. Blows were delivered to his legs, arms, back, head... nothing was safe from the evil, spreading like the plague, it was a plague, one after the other he could feel the cane cutting into him, beating him, testing him. His body moved violently with the force of the blows. His head was buried in his arms on the stone pillar, holding tightly, for dear life. The pillar was red with the blood of many a tortured victim. His blood, was now being shared as he humbled himself to be ordinary, to be mortal.
The blows suddenly stopped, he did not here the order. He rested momentarily, this moment, almost a blessing, he began blocking out the noise of the world, silence and the beating of his heart, was all he heard now, pumping life, to be taken away. He reflected on the past thirty or so years of his life, searching for answers and reasons. He walked to many a place and he remembered all with crystal clarity but he had favourites. Certain trees, certain stones, certain sentiments... certain smells and certain dream lives. His life was never easy but always desired. Waking up, the best part. The morning sun, the evening dew. The silent mystery of the night, as he lay, ready to dream, under the guidance of his Father. Hushed whispers of children laughing. It was going away... slow. The noises returned. He fought with his legs, slowly stood up, battling with himself to stand strong, blood and sweat dripping off him. His face was now closer to the sky, to heaven. The soldiers were exhausted, from heavy drinking but were forced to make a weapon change. “My heart is ready Father, my heart is ready...” were the words he spoke, as he stood, beaten and in such pain. He was prepared to undertake the second stage of his slaughter, the stage that would share him. The soldiers were testing their weapon of choice on the nearby table. Nine tails. Nine pieces of cotton cord with pieces of sharp metal and bone attached to the end. It was designed not just to torture, but to kill. Despite their exhaustion, the eagerness to use this weapon of death was ever so apparent, like a child with a new toy. He waited, painfully, patiently, for the first order. It came.
UNUS... the tails cut through him like butter, he clenched his hands and almost fell over, back arched forward. He could feel the blood leaking down his back. His eyes closed tightly, an unbearable pain, an unbearable love. DUO... his right shoulder ripped by the shards, he yelled a whisper of pain as he watched fragments fall by him. He almost fell to his knees, hands desperately held him up, shaking uncontrollably once more, tears in his eyes, the pain: unbounded. His passion and love: unbounded once more. TRES... he could feel his own blood hit his hair and neck as the shards tore through his back. His legs fail and he once again fell, his knees making a collision with the hard stone floor. QUATTOUR... a blow to the right side of his body, blood and flesh dispensed around him, his whole body was shaking, his hands could not grip, could not clench, they were desperately trying to hold onto the pillar as he hung off it. His bones could feel the cold air... SEX... he could see his own blood slowly painting the floor a bright red. These blows were not just cutting into him, but beating into him, the power of the blows, slowly killing him. His loving mother couldn’t bare the pain of her most loving son being put through this. SEPTEM... a blood cloud as the sound of the knock echoed through the Preatorium... the soldiers were roaring with laughter, and screaming as they knocked him, again and again and again, relentless, violent, evil. His arms were gripped around the pillar, holding on for dear life, all the while feeling the skin and muscles being torn and ripped to the bone, by the bone. His arms and legs were being hacked apart, each blow sounding more and more liquefied... QUINDECIM... the blow was delivered to his head and face, pulling hair and skin off. He was hanging now, unable to hold onto anything... he was beginning to no longer look human. He was burning and was now less than ordinary. The Master brought down to the level of a criminal, still as he hung there, more and more blows were delivered. The Master. Everywhere, he tried to spread a word, some truth, about this mysterious and barbaric thing called life. To spread the idea of love and humanity, of forgiveness and compassion, of pity and pleasure. The preservation of nature. A Prince of Peace.
He was being butchered. His voice could barely operate, and so his screams were not heard, except by his Father... UNDEVIGINTI... the sound of scattered blood... VIGINTI UNUS... bone against bone... VIGINTI NOVEM... his structure cracks and he exclaims a roar of utter agony... TRIGINTA... he could feel himself suffocating as his chest tightened... TRIGINTA UNUS... the tails now glided through him... TRIGINTA DUO... blood, flesh and muscle dispersed once more... a blood cloud...
... TRIGINTA TRES...
... TRIGINTA QUATTOR...
... TRIGINTA QUINQUE...
... TRIGINTA SEX... blood clouds...
... TRIGINTA SEPTEM...
... TRIGINTA OCTO... his screams were now breathless whispers...
... TRIGINTA NOVEM...
The blows stopped temporarily. Is it over? Footsteps and laughter as the monsters unlocked the chains on one of his hands, and flipped him over onto his back. He fell into a pool of red liquid, belly exposed. The liquid cooled him as he lay there, barely alive. He could see their faces, he could see them, through blood-stained eyes. The soldiers too were covered with his blood, as they created this holy bloodbath, but their eyes were filled with darkness and were coldly black. He meant nothing to them, while they meant everything to him. He loved them despite the fact they were killing him. He lay there, with his head tilted slightly to the left, and could see the entrance to the Praetorium. He could feel his cooled blood under him, smelled its metallic odour and tasted the results of his suffering, and he lowered his head again, and saw pieces of his body, lying all around him... flesh and blood. He could barely move, knowing that it would begin again, still the beginning of his journey to save the world. He could no longer see his mother. Even now, his thoughts lingered on her. Memories, of playing in the yard as a child, being cared for, being put to sleep. Bedtime stories. Night time stars.
The soldiers began delivering more blows to his front side, attacking his stomach, chest, arms and his legs...
He could not scream, he could not move, he lay there, watching his own blood and skin being scattered on the soldiers, the floor and his own face, as if he could not feel the pain, such was the extent of it. The pain of deceit, betrayal, lies and power, his heart felt this pain, his heart: dying for and from this pain. He could see the blows being delivered, and the results of those blows, leaving the putrid marks on his weak and innocent body. The tails opening his body and digging deeply into his flesh and muscle, and pulling and ripping and dispensing, solid pieces of meat, being removed... a butchering. The blood was leaking from his sides, gushing, the tears leaking from his eyes, crying. How powerful he was, to live a life knowing that this would be his fate, his end. The blows to his legs were powerful enough to move as well as lacerate them, causing extreme strain to his upper body, his eyes were the entrance to a seen pain. He could feel his death coming and the life slowly leaving him, the time coming where the suffering would end. The liquid beating, the sounds of tearing skin and flesh, the sounds of laughter, the sounds of sandals on blood. The life was leaving him, slowly, gently, as the blows were quick, brutal.
The blows stopped. The Centurion had entered, shocked at what he saw laying there in the river of helplessness. He screamed an order and they unlocked his last chain and he lay there, barely able to move. He looked up into the blue sky. A soft quiet beauty and a sense of silence and peace prevailed in that deep, blue and infinite sky. A dove, blindingly white, flew above him, almost in slow motion. He yearned and he cried.
He felt a sharp sudden pain as he realised they were dragging him slowly out, his skinless back being rubbed against the stone floor, leaving a trail of red evil. Red, thick islands on a red lake, remained. A red river.
He closed his blood-stained, tearful eyes as he was taken away... and prepared.
A love story, between God and man. This love story, it had begun...
Byron P. Brassel
Holy Week - 2009
Re-edited and re-mastered - November 2010, April 2011, June 2012
He could hear them, monsters holding tools of pain, preparing, reveling, to use the results of their wicked inventions. Yet he began praying for them, asking for forgiveness for their sins. The silent, powerful presence he evoked was ever so evident, yet he said nothing... he did not need to. His silence now, was his strength. The moment was drawing near; the soldiers were in position, waiting for the signal, the signal that would unleash the pain of the world. It came.
Unus... He could feel the pain slither up his spine, as the cane slapped his back... claw its way up his neck and in his heart, a ruthless, evil, searing pain. He made no cry, his face told the story, his eyes closed tight, clenched teeth, waiting... Duo... the pain doubled, his legs wet with a painful sweat, his back was arched, he couldn’t move and the cold breeze cut through the valley of his scars. Tres... he clenched his fists, the searing pain travelled through him, the journey to save the world, begun, how great is his love, that he would endure the torture, the pain, the suffering... how great is his love, that he would willingly walk into the shadow of all evil, to save a forsaken world. Blow after blow he received from the cane, the slapping noise echoed across the Preatorium, his eyes, wet with the tears, his hands shaking, his body shaking, his legs, barely able to hold him up. More blows were delivered like rain drops falling from the clouds... his legs gave way, his weak knees land on the cold stone floor, he could feel his open scars screaming the pain of the world, a forsaken pain. Blows were delivered to his legs, arms, back, head... nothing was safe from the evil, spreading like the plague, it was a plague, one after the other he could feel the cane cutting into him, beating him, testing him. His body moved violently with the force of the blows. His head was buried in his arms on the stone pillar, holding tightly, for dear life. The pillar was red with the blood of many a tortured victim. His blood, was now being shared as he humbled himself to be ordinary, to be mortal.
The blows suddenly stopped, he did not here the order. He rested momentarily, this moment, almost a blessing, he began blocking out the noise of the world, silence and the beating of his heart, was all he heard now, pumping life, to be taken away. He reflected on the past thirty or so years of his life, searching for answers and reasons. He walked to many a place and he remembered all with crystal clarity but he had favourites. Certain trees, certain stones, certain sentiments... certain smells and certain dream lives. His life was never easy but always desired. Waking up, the best part. The morning sun, the evening dew. The silent mystery of the night, as he lay, ready to dream, under the guidance of his Father. Hushed whispers of children laughing. It was going away... slow. The noises returned. He fought with his legs, slowly stood up, battling with himself to stand strong, blood and sweat dripping off him. His face was now closer to the sky, to heaven. The soldiers were exhausted, from heavy drinking but were forced to make a weapon change. “My heart is ready Father, my heart is ready...” were the words he spoke, as he stood, beaten and in such pain. He was prepared to undertake the second stage of his slaughter, the stage that would share him. The soldiers were testing their weapon of choice on the nearby table. Nine tails. Nine pieces of cotton cord with pieces of sharp metal and bone attached to the end. It was designed not just to torture, but to kill. Despite their exhaustion, the eagerness to use this weapon of death was ever so apparent, like a child with a new toy. He waited, painfully, patiently, for the first order. It came.
UNUS... the tails cut through him like butter, he clenched his hands and almost fell over, back arched forward. He could feel the blood leaking down his back. His eyes closed tightly, an unbearable pain, an unbearable love. DUO... his right shoulder ripped by the shards, he yelled a whisper of pain as he watched fragments fall by him. He almost fell to his knees, hands desperately held him up, shaking uncontrollably once more, tears in his eyes, the pain: unbounded. His passion and love: unbounded once more. TRES... he could feel his own blood hit his hair and neck as the shards tore through his back. His legs fail and he once again fell, his knees making a collision with the hard stone floor. QUATTOUR... a blow to the right side of his body, blood and flesh dispensed around him, his whole body was shaking, his hands could not grip, could not clench, they were desperately trying to hold onto the pillar as he hung off it. His bones could feel the cold air... SEX... he could see his own blood slowly painting the floor a bright red. These blows were not just cutting into him, but beating into him, the power of the blows, slowly killing him. His loving mother couldn’t bare the pain of her most loving son being put through this. SEPTEM... a blood cloud as the sound of the knock echoed through the Preatorium... the soldiers were roaring with laughter, and screaming as they knocked him, again and again and again, relentless, violent, evil. His arms were gripped around the pillar, holding on for dear life, all the while feeling the skin and muscles being torn and ripped to the bone, by the bone. His arms and legs were being hacked apart, each blow sounding more and more liquefied... QUINDECIM... the blow was delivered to his head and face, pulling hair and skin off. He was hanging now, unable to hold onto anything... he was beginning to no longer look human. He was burning and was now less than ordinary. The Master brought down to the level of a criminal, still as he hung there, more and more blows were delivered. The Master. Everywhere, he tried to spread a word, some truth, about this mysterious and barbaric thing called life. To spread the idea of love and humanity, of forgiveness and compassion, of pity and pleasure. The preservation of nature. A Prince of Peace.
He was being butchered. His voice could barely operate, and so his screams were not heard, except by his Father... UNDEVIGINTI... the sound of scattered blood... VIGINTI UNUS... bone against bone... VIGINTI NOVEM... his structure cracks and he exclaims a roar of utter agony... TRIGINTA... he could feel himself suffocating as his chest tightened... TRIGINTA UNUS... the tails now glided through him... TRIGINTA DUO... blood, flesh and muscle dispersed once more... a blood cloud...
... TRIGINTA TRES...
... TRIGINTA QUATTOR...
... TRIGINTA QUINQUE...
... TRIGINTA SEX... blood clouds...
... TRIGINTA SEPTEM...
... TRIGINTA OCTO... his screams were now breathless whispers...
... TRIGINTA NOVEM...
The blows stopped temporarily. Is it over? Footsteps and laughter as the monsters unlocked the chains on one of his hands, and flipped him over onto his back. He fell into a pool of red liquid, belly exposed. The liquid cooled him as he lay there, barely alive. He could see their faces, he could see them, through blood-stained eyes. The soldiers too were covered with his blood, as they created this holy bloodbath, but their eyes were filled with darkness and were coldly black. He meant nothing to them, while they meant everything to him. He loved them despite the fact they were killing him. He lay there, with his head tilted slightly to the left, and could see the entrance to the Praetorium. He could feel his cooled blood under him, smelled its metallic odour and tasted the results of his suffering, and he lowered his head again, and saw pieces of his body, lying all around him... flesh and blood. He could barely move, knowing that it would begin again, still the beginning of his journey to save the world. He could no longer see his mother. Even now, his thoughts lingered on her. Memories, of playing in the yard as a child, being cared for, being put to sleep. Bedtime stories. Night time stars.
The soldiers began delivering more blows to his front side, attacking his stomach, chest, arms and his legs...
He could not scream, he could not move, he lay there, watching his own blood and skin being scattered on the soldiers, the floor and his own face, as if he could not feel the pain, such was the extent of it. The pain of deceit, betrayal, lies and power, his heart felt this pain, his heart: dying for and from this pain. He could see the blows being delivered, and the results of those blows, leaving the putrid marks on his weak and innocent body. The tails opening his body and digging deeply into his flesh and muscle, and pulling and ripping and dispensing, solid pieces of meat, being removed... a butchering. The blood was leaking from his sides, gushing, the tears leaking from his eyes, crying. How powerful he was, to live a life knowing that this would be his fate, his end. The blows to his legs were powerful enough to move as well as lacerate them, causing extreme strain to his upper body, his eyes were the entrance to a seen pain. He could feel his death coming and the life slowly leaving him, the time coming where the suffering would end. The liquid beating, the sounds of tearing skin and flesh, the sounds of laughter, the sounds of sandals on blood. The life was leaving him, slowly, gently, as the blows were quick, brutal.
The blows stopped. The Centurion had entered, shocked at what he saw laying there in the river of helplessness. He screamed an order and they unlocked his last chain and he lay there, barely able to move. He looked up into the blue sky. A soft quiet beauty and a sense of silence and peace prevailed in that deep, blue and infinite sky. A dove, blindingly white, flew above him, almost in slow motion. He yearned and he cried.
He felt a sharp sudden pain as he realised they were dragging him slowly out, his skinless back being rubbed against the stone floor, leaving a trail of red evil. Red, thick islands on a red lake, remained. A red river.
He closed his blood-stained, tearful eyes as he was taken away... and prepared.
A love story, between God and man. This love story, it had begun...
Byron P. Brassel
Holy Week - 2009
Re-edited and re-mastered - November 2010, April 2011, June 2012
Sunday, June 10, 2012
Sylow's Theorem
This post is not just a statement and proof of a theorem in mathematics, like I usually tend to do here. It is something slightly more powerful. This is a dedication, to a man who has provided my life with a proof and result that lies right up there with my favourite things in this world. A result so seemingly complete in its statements, arguments and orchestrations, it seems to test the very fabric of the word 'perfect'.
First, about the man. Peter Ludwig Mejdell Sylow (12 December 1832 - 7 September 1918) was a Norwegian mathematician who did, or should I more gracefully say, pioneered, in work relating to finite group theory. He composed what we call the Sylow Theorems which deal with the number of 'fixed-ordered' subgroups that any given finite group contains.
Now to clear some things up. For those of you who do not know what a group is, refer to my last post about Group Theory and the Rubik's Cube. The proof you see there is, unusually, numbered in order to make its statements more legible to the layman, and perhaps, add to the aesthetics. However, I will define it now, anyway. A group in mathematics, is just an algebraic structure that has a set under some operation, could be addition, or multiplication, where any two elements can form a third element. In order for any algebraic structure to qualify as a group it must first adhere to the so called group axioms, i.e., the set must be closed under its operation, it must have an identity, every element must have an inverse element, and the operation must be associative. If any of these fail, the structure will not be a group (perhaps something weaker like a 'semi-group'). Now a finite group is merely a group with a countable amount of elements, and when one speaks of an 'order', one merely means this: the order of a group is the number of elements that group has, also called its cardinality. Finally, to define a subgroup... it is probably what you expect it to be. A subgroup is a set under the same operation as the original 'parent' group, that preserves all the axioms, and is a SUBSET of the set in the parent group. i.e., it is a smaller group inside the parent group.
Alright, so the small lesson is over. Now to get to the juicy bits of this post. It must first be noted that in line 9 of this post I mentioned that Sylow had theorems. Yes, more than one. I will be proving what is called Theorem 1. The others are smaller results, and I shant get into them here (You can either ask me or look them up yourselves). Now, what makes this theorem so special? Why have I seemingly come here to waste time in writing something about it? Why does it nudge perfection?
One answer is that its proof uses so many of the techniques of group theory in its sequential steps that it almost seems to have been orchestrated by God. When I read this proof, or indeed, write it, I can almost hear it, like a piece of music, and this happens to me a lot. I look at mathematics like music. If I write the third line of my fourth proof any differently, will it sound different? Maybe replace an oboe with a bassoon, perhaps change the note from E-flat major on strings to B flat? The answer is yes. Writing a proof, to me, is like writing a score. You sit at a desk and you doodle, you play around with ideas, you write a draft. You have an idea in your head as to what it will sound like, or how strong it is. You then take the score to the players, and have them perform it. In mathematics, this is the same. A proof is always looked at by others in order to do one thing, seek perfection. When the orchestra performs a 'first-draft' cue, quite often, not always, imperfections are heard (of course ignoring performance errors). The whole idea of a perfect proof, is this: It must be as short and as simple as possible. That is all.
We call this, elegance. Quite often in music, orchestral music, which I know well, the French horn players, for example, will play a motif, two, three times over, followed by a layer of strings playing a motif that induces a counterpoint, for example, but upon listening and reflecting, one may realise that playing it two repeated times makes the piece far superior a listening experience, rather than repeating it a third time. I hope you can forgive my constant comparisons to music, but I never want to be seen as someone who writes mathematics, but rather, as someone who orchestrates it. When reading Sylow's Theorem, I can almost hear a perfect piece of music being played just to me, and people often ask me why I love mathematics so much. This is why. This kind of thing, seeing it from this kind of perspective, makes me feel alive. I exist to witness such moments.
I will try and make everything in this proof as understandable as possible, so if anything is confusing and I know it will be, you can get back to me if it really bugs you, so, just bear with me, yes?
Statement: Let G be a finite group and p, α ∈ ℕ, where p is prime. If p^α | o(G) [p divides o(G)], then G has a subgroup of order p^α.
Proof: The proof is by induction on o(G), the order of the group. If o(G) is 1 or 2, the result is clearly true. So now I assume that o(G) > 2 and the result is true for all groups of order less than o(G). This is the Induction Hypothesis. Now I assume that p^α | o(G). Let o(G) = γ_0 + ∑ γ_i (where i runs from 1 to s) be the class equation of G (so γ_0 = o(Z_G) where Z_G is the centre of the group, and γ_i = |C_i| for i = 1, ..., s, where C_1, ..., C_s are the distinct conjugacy classes of G that have more than one element).
I consider two cases now.
CASE 1: Suppose p ⫮ γ_0. Then, since p | o(G), there exist j = 1, ..., s such that p ⫮ γ_j (otherwise p would divide o(G) - ∑ γ_i = γ_0 which is a contradiction). Since p ⫮ γ_j and p is prime, we have (p, γ_j) = 1. It follows that (p^α, γ_j) = 1. Now let x ∈ C_j, so C_j = (x) and |(x)| = |C_j| = γ_j ≥ 2. So o(G) = o(Z_x)⋅γ_j ... (i). Since p^α | o(G), i.e., p^α | o(Z_x)⋅γ_j, and (p^α, γ_j) = 1, it follows that p^α | o(Z_x) ... (ii). Now, in (i), γ_j ≥ 2, so o(Z_x) < o(G). By the Induction Hypothesis and (ii), Z_x must have a subgroup H with o(H) = p^α. Now H < Z_x [H is a subgroup of Z_x] and Z_x < G, so H < G and o(H) = p^α, as required.
CASE 2: Suppose p | γ_0, i.e., p | o(Z_G). Since Z_G is an abelian group (a group where elements are commutative too), it follows that there exists an element, a in Z_G such that o(a) = p. Now 〈a〉 = {e, a, a^2, ..., a^p-1} ⊆ Z_G, so (a^i) = {a^i} for i = 0, 1, 2, ..., p-1. Thus, 〈a〉 is a union of one-element conjugacy classes of G, so 〈a〉 ⊲ G [〈a〉 is a normal subgroup of G]. (A normal subgroup, H, say, is just a subgroup that is invariant under its conjugation, i.e., for an element a ∈ H, and an element g ∈ G, if we have gag^-1 ⊆ H, then H is a normal subgroup of G). (Alright, so now I will define something called a homomorphism, but first, what the hell is a homomorphism? It is merely a structure-preserving map from one group, in this case, to another group. If a function is a homomorphism from G to H (not necessarily G's subgroup, but any group), then for any two elements g_1, g_2 in the group G, under the operation *, say, f(g_1 * g_2) = f(g_1) * f(g_2). If this function is also surjective (onto), then we can say that G is isomorphic to H, i.e., It is structurally identical to H.). Okey dokey, let η : G ⟶ G/〈a〉 be the canonical (onto) homomorphism. Note that "/" in this case does not mean 'divide', it means 'G factored by 〈a〉'. η(g) = 〈a〉g where g is an element of G. Then ker η = 〈a〉, and o(G/〈a〉) = o(G)/o(〈a〉) = o(G)/p < o(G) ... (iii) [The kernel of any given mapping is just the set of elements that are mapped to the identity or zero element]. Now, p^α | o(G), so p^α/p | o(G)/p, i.e., p^α-1 | o(G/〈a〉) ... (iv). By (iii), (iv) and the Induction Hypothesis, there exists B < G/〈a〉 such that o(B) = p^α-1. Now, define A = {g ∈ G : η(g) ∈B}. Then, A = {g ∈ G : 〈a〉g ∈B} and A < G, and the restriction η|_A of η to the group A is a group homomorphism whose image is B (by the definition of A). Note that 〈a〉 ⊆ A; if g ∈ 〈a〉, then η(g) = 〈a〉g = 〈a〉e ∈B, so g ∈A. Thus, ker(η|_A) = A ⋂ ker η = A ⋂ 〈a〉 = 〈a〉. Applying the homomorphism theorem to η|_A : A ⟶ B, since ker(η|_A) = 〈a〉, we have that the domain of η|_A, factored by the kernel of η|_A is isomorphic to the image of η|_A, i.e., A/〈a〉 is isomorphic to B. So, o(A)/p = o(G/〈a〉) = o(B) = p^α-1, hence, o(A) = pp^α-1 = p^α. Since A < G, the proof is now completed. ∎
Byron P. Brassel
2012/06/11
Sources:
Groups - J. G. Raftery, Ph.D, University Of Natal (Westville)
Wikipedia pages on Groups, subgroups, finiteness and mappings.
First, about the man. Peter Ludwig Mejdell Sylow (12 December 1832 - 7 September 1918) was a Norwegian mathematician who did, or should I more gracefully say, pioneered, in work relating to finite group theory. He composed what we call the Sylow Theorems which deal with the number of 'fixed-ordered' subgroups that any given finite group contains.
Now to clear some things up. For those of you who do not know what a group is, refer to my last post about Group Theory and the Rubik's Cube. The proof you see there is, unusually, numbered in order to make its statements more legible to the layman, and perhaps, add to the aesthetics. However, I will define it now, anyway. A group in mathematics, is just an algebraic structure that has a set under some operation, could be addition, or multiplication, where any two elements can form a third element. In order for any algebraic structure to qualify as a group it must first adhere to the so called group axioms, i.e., the set must be closed under its operation, it must have an identity, every element must have an inverse element, and the operation must be associative. If any of these fail, the structure will not be a group (perhaps something weaker like a 'semi-group'). Now a finite group is merely a group with a countable amount of elements, and when one speaks of an 'order', one merely means this: the order of a group is the number of elements that group has, also called its cardinality. Finally, to define a subgroup... it is probably what you expect it to be. A subgroup is a set under the same operation as the original 'parent' group, that preserves all the axioms, and is a SUBSET of the set in the parent group. i.e., it is a smaller group inside the parent group.
Alright, so the small lesson is over. Now to get to the juicy bits of this post. It must first be noted that in line 9 of this post I mentioned that Sylow had theorems. Yes, more than one. I will be proving what is called Theorem 1. The others are smaller results, and I shant get into them here (You can either ask me or look them up yourselves). Now, what makes this theorem so special? Why have I seemingly come here to waste time in writing something about it? Why does it nudge perfection?
One answer is that its proof uses so many of the techniques of group theory in its sequential steps that it almost seems to have been orchestrated by God. When I read this proof, or indeed, write it, I can almost hear it, like a piece of music, and this happens to me a lot. I look at mathematics like music. If I write the third line of my fourth proof any differently, will it sound different? Maybe replace an oboe with a bassoon, perhaps change the note from E-flat major on strings to B flat? The answer is yes. Writing a proof, to me, is like writing a score. You sit at a desk and you doodle, you play around with ideas, you write a draft. You have an idea in your head as to what it will sound like, or how strong it is. You then take the score to the players, and have them perform it. In mathematics, this is the same. A proof is always looked at by others in order to do one thing, seek perfection. When the orchestra performs a 'first-draft' cue, quite often, not always, imperfections are heard (of course ignoring performance errors). The whole idea of a perfect proof, is this: It must be as short and as simple as possible. That is all.
We call this, elegance. Quite often in music, orchestral music, which I know well, the French horn players, for example, will play a motif, two, three times over, followed by a layer of strings playing a motif that induces a counterpoint, for example, but upon listening and reflecting, one may realise that playing it two repeated times makes the piece far superior a listening experience, rather than repeating it a third time. I hope you can forgive my constant comparisons to music, but I never want to be seen as someone who writes mathematics, but rather, as someone who orchestrates it. When reading Sylow's Theorem, I can almost hear a perfect piece of music being played just to me, and people often ask me why I love mathematics so much. This is why. This kind of thing, seeing it from this kind of perspective, makes me feel alive. I exist to witness such moments.
I will try and make everything in this proof as understandable as possible, so if anything is confusing and I know it will be, you can get back to me if it really bugs you, so, just bear with me, yes?
Statement: Let G be a finite group and p, α ∈ ℕ, where p is prime. If p^α | o(G) [p divides o(G)], then G has a subgroup of order p^α.
Proof: The proof is by induction on o(G), the order of the group. If o(G) is 1 or 2, the result is clearly true. So now I assume that o(G) > 2 and the result is true for all groups of order less than o(G). This is the Induction Hypothesis. Now I assume that p^α | o(G). Let o(G) = γ_0 + ∑ γ_i (where i runs from 1 to s) be the class equation of G (so γ_0 = o(Z_G) where Z_G is the centre of the group, and γ_i = |C_i| for i = 1, ..., s, where C_1, ..., C_s are the distinct conjugacy classes of G that have more than one element).
I consider two cases now.
CASE 1: Suppose p ⫮ γ_0. Then, since p | o(G), there exist j = 1, ..., s such that p ⫮ γ_j (otherwise p would divide o(G) - ∑ γ_i = γ_0 which is a contradiction). Since p ⫮ γ_j and p is prime, we have (p, γ_j) = 1. It follows that (p^α, γ_j) = 1. Now let x ∈ C_j, so C_j = (x) and |(x)| = |C_j| = γ_j ≥ 2. So o(G) = o(Z_x)⋅γ_j ... (i). Since p^α | o(G), i.e., p^α | o(Z_x)⋅γ_j, and (p^α, γ_j) = 1, it follows that p^α | o(Z_x) ... (ii). Now, in (i), γ_j ≥ 2, so o(Z_x) < o(G). By the Induction Hypothesis and (ii), Z_x must have a subgroup H with o(H) = p^α. Now H < Z_x [H is a subgroup of Z_x] and Z_x < G, so H < G and o(H) = p^α, as required.
CASE 2: Suppose p | γ_0, i.e., p | o(Z_G). Since Z_G is an abelian group (a group where elements are commutative too), it follows that there exists an element, a in Z_G such that o(a) = p. Now 〈a〉 = {e, a, a^2, ..., a^p-1} ⊆ Z_G, so (a^i) = {a^i} for i = 0, 1, 2, ..., p-1. Thus, 〈a〉 is a union of one-element conjugacy classes of G, so 〈a〉 ⊲ G [〈a〉 is a normal subgroup of G]. (A normal subgroup, H, say, is just a subgroup that is invariant under its conjugation, i.e., for an element a ∈ H, and an element g ∈ G, if we have gag^-1 ⊆ H, then H is a normal subgroup of G). (Alright, so now I will define something called a homomorphism, but first, what the hell is a homomorphism? It is merely a structure-preserving map from one group, in this case, to another group. If a function is a homomorphism from G to H (not necessarily G's subgroup, but any group), then for any two elements g_1, g_2 in the group G, under the operation *, say, f(g_1 * g_2) = f(g_1) * f(g_2). If this function is also surjective (onto), then we can say that G is isomorphic to H, i.e., It is structurally identical to H.). Okey dokey, let η : G ⟶ G/〈a〉 be the canonical (onto) homomorphism. Note that "/" in this case does not mean 'divide', it means 'G factored by 〈a〉'. η(g) = 〈a〉g where g is an element of G. Then ker η = 〈a〉, and o(G/〈a〉) = o(G)/o(〈a〉) = o(G)/p < o(G) ... (iii) [The kernel of any given mapping is just the set of elements that are mapped to the identity or zero element]. Now, p^α | o(G), so p^α/p | o(G)/p, i.e., p^α-1 | o(G/〈a〉) ... (iv). By (iii), (iv) and the Induction Hypothesis, there exists B < G/〈a〉 such that o(B) = p^α-1. Now, define A = {g ∈ G : η(g) ∈B}. Then, A = {g ∈ G : 〈a〉g ∈B} and A < G, and the restriction η|_A of η to the group A is a group homomorphism whose image is B (by the definition of A). Note that 〈a〉 ⊆ A; if g ∈ 〈a〉, then η(g) = 〈a〉g = 〈a〉e ∈B, so g ∈A. Thus, ker(η|_A) = A ⋂ ker η = A ⋂ 〈a〉 = 〈a〉. Applying the homomorphism theorem to η|_A : A ⟶ B, since ker(η|_A) = 〈a〉, we have that the domain of η|_A, factored by the kernel of η|_A is isomorphic to the image of η|_A, i.e., A/〈a〉 is isomorphic to B. So, o(A)/p = o(G/〈a〉) = o(B) = p^α-1, hence, o(A) = pp^α-1 = p^α. Since A < G, the proof is now completed. ∎
Byron P. Brassel
2012/06/11
Sources:
Groups - J. G. Raftery, Ph.D, University Of Natal (Westville)
Wikipedia pages on Groups, subgroups, finiteness and mappings.
Tuesday, June 5, 2012
Group Theory and The Rubik's Cube
Let (G ; ∗) be
the set of movements of the Rubik's Cube. A movement is defined as follows; one possible
move is a clockwise turn of the top face followed by a counter-clockwise turn
of the right face. If M_1 and M_2 are two moves, then M_1 ∗ M_2 is the move where one does M_1 first, and
then M_2.
PROVE that the set (G ; ∗) is indeed a group under the operation of ∗.
PROVE that the set (G ; ∗) is indeed a group under the operation of ∗.
Proof: I shall now prove this using the axioms of group theory.
(1) It is firstly required
that ∗ be well-defined. It is easy to see this, for if M_1 and M_2
are moves, M_1 ∗ M_2 is a move as well. So G is closed under ∗.
(2) Now, let e be the 'empty' move, i.e. a move that does not change the configuration of the cube in any way. Therefore, M ∗ e means 'first do M, then do nothing'. Therefore, M ∗ e = M. So we have proven that G has an identity, namely, e.
(3) To prove G has an inverse is not taxing in the least. Since M is a move, we can reverse this movement, i.e. do the steps backward to get M', say. Then M ∗ M' simply means 'do M, then reverse the steps of M', i.e. M ∗ M' = e, the empty move. Thus, M' is the inverse of M. Hence it is proved that every element of G has an inverse.
(4) To prove associativity requires a bit more thought and intuition, as well as mechanical sympathy for the structure of the Rubik's cube itself. Upon reflection, and gratuitous analysis, one can see that the cube is made up of 27 smaller cubes of which one cannot be seen. These 'mini' cubes shall be called 'Cubies' for convenience. The space in which these cubies occupy shall be called cubicles. If C is the oriented cubie, write M(C) for the oriented cubicle that C ends up in after the move M is applied, with the faces of M(C) written in the same order as the faces of C. Now, the move M_1 moves C to the cubicle M_1(C); the move M_2 moves M_1(C) to M_2(M_1(C)). Therefore, (M_1 ∗ M_2)(C) = M_2(M_1(C)).
[Recall how moves are defined in the statement above if you are confused]
It is now required that (M_1 ∗ M_2) ∗ M_3 = M_1 ∗ (M_2 ∗ M_3)... (i) be proved, for moves M_1, M_2 and M_3. Upon completion of this, we will have associativity, and the result shall follow... great times.
The proof of (i) above is a direct consequence of proving that [(M_1 ∗ M_2) ∗ M_3](C) = [M_1 ∗ (M_2 ∗ M_3)](C), for any cubie C. Now, [(M_1 ∗ M_2) ∗ M_3](C) = M_3([M_1 ∗ M_2](C)) = M_3(M_2(M_1(C))).
Also, [M_1 ∗ (M_2 ∗ M_3)](C) = (M_2 ∗ M_3)(M_1(C)) = M_3(M_2(M_1(C))).
So (M_1 ∗ M_2) ∗ M_3 = M_1 ∗ (M_2 ∗ M_3). Therefore ∗ is associative.
Thus, finally, by (1), (2), (3) and (4), (G ; ∗) is a group under ∗. ∎
(2) Now, let e be the 'empty' move, i.e. a move that does not change the configuration of the cube in any way. Therefore, M ∗ e means 'first do M, then do nothing'. Therefore, M ∗ e = M. So we have proven that G has an identity, namely, e.
(3) To prove G has an inverse is not taxing in the least. Since M is a move, we can reverse this movement, i.e. do the steps backward to get M', say. Then M ∗ M' simply means 'do M, then reverse the steps of M', i.e. M ∗ M' = e, the empty move. Thus, M' is the inverse of M. Hence it is proved that every element of G has an inverse.
(4) To prove associativity requires a bit more thought and intuition, as well as mechanical sympathy for the structure of the Rubik's cube itself. Upon reflection, and gratuitous analysis, one can see that the cube is made up of 27 smaller cubes of which one cannot be seen. These 'mini' cubes shall be called 'Cubies' for convenience. The space in which these cubies occupy shall be called cubicles. If C is the oriented cubie, write M(C) for the oriented cubicle that C ends up in after the move M is applied, with the faces of M(C) written in the same order as the faces of C. Now, the move M_1 moves C to the cubicle M_1(C); the move M_2 moves M_1(C) to M_2(M_1(C)). Therefore, (M_1 ∗ M_2)(C) = M_2(M_1(C)).
[Recall how moves are defined in the statement above if you are confused]
It is now required that (M_1 ∗ M_2) ∗ M_3 = M_1 ∗ (M_2 ∗ M_3)... (i) be proved, for moves M_1, M_2 and M_3. Upon completion of this, we will have associativity, and the result shall follow... great times.
The proof of (i) above is a direct consequence of proving that [(M_1 ∗ M_2) ∗ M_3](C) = [M_1 ∗ (M_2 ∗ M_3)](C), for any cubie C. Now, [(M_1 ∗ M_2) ∗ M_3](C) = M_3([M_1 ∗ M_2](C)) = M_3(M_2(M_1(C))).
Also, [M_1 ∗ (M_2 ∗ M_3)](C) = (M_2 ∗ M_3)(M_1(C)) = M_3(M_2(M_1(C))).
So (M_1 ∗ M_2) ∗ M_3 = M_1 ∗ (M_2 ∗ M_3). Therefore ∗ is associative.
Thus, finally, by (1), (2), (3) and (4), (G ; ∗) is a group under ∗. ∎
Sunday, April 22, 2012
Let X equal the quantity of all quantities of X
Let X equal the quantity of all quantities of X. Let X equal the cold.
The months of cold equal May through August. There are four months of
cold, and four of heat, leaving four months of indeterminate
temperature. In June it snows in Drakensberg. In February and July the
students return and Adams bookshop, in Westville campus (Ukzn) is full.
Adams coffee bar is full from February through June and July through
November. Two months of heat, four months of cold and four months of
indeterminate form. Let X equal the month when Adams bookshop is full.
The number of books approaches infinity as the number of months of cold
approaches four. Let X equal the quantity of months when Adam's coffee
bar is open. The number of dishes sold approaches infinity as the number
of dishes produced approaches infinity as the number of books sold in
the one month approaches infinity as the number of months of cold
approaches four. I will be colder in the future than I am now. The
future of cold is infinite. The future of heat is the future of cold.
Adams bookshop is infinite and so is never full except in February and
July and the coffee bar will never be full except from February through
June and July through November...
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