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July 9, 2010Teorema Isomorfisma Pertama

By Aria Turns ¶ Posted in aljabar abstrak, pembuktian ¶ Tagged isomorfisma, matematika, Math, pertama, teorema

¶ 9 Comments

Okey kita lupakan sejenak apakah Teorema Fundamental Homomorfisma ( (2010/07/04/dua-teorema-yang-sama-

kah/)The Fundamental homomorphism Theorem (2010/07/04/dua-teorema-yang-sama-kah/)) sama dengan Teorema

Isomorfisma Pertama ( (2010/07/04/dua-teorema-yang-sama-kah/) First Isomorphism Theorem (2010/07 /04/dua-teorema-

yang-sama-kah/)). (2010/07/04/dua-teorema-yang-sama-kah/) Kali ini saya mau membahas Teorema Isomorfisma

Pertama, Lho kenapa bukan Teorema Fundamental Homomorfisma? Karena dari semua literatur aljabar yan saya baca

tidak terdapat perbedaan, semuanya satu suara mengenai Teorema Isomorfisma Pertama

Diberikan Homomorfisma grup maka akan terbentuk

1. kernel dari

2. image dari , dengan mudah dapat kita tunjukan merupakan subgrup dari

3. Karena kernel merupakan subgrup normal maka terbentuk grup faktor

Teorema Isomorfisma Pertama mengatakan bahwa dan isomorfis.

Teorema Isomorfisma Pertama: Diberikan Homomorfisma grup dengan kernel dan homomorfisma

natura (http:/ /mathworld.wolfram.com/NaturalHomomorphism.html)l maka terdapat isomorfisma

.

Jika dibuatkan diagaramnya maka teorema tersebut akan tampak sebagai berikut

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Bukti:

KIta definisikan , dengan . Pertama-tama kita tunjukan well-defined

(http://mathworld.wolfram.com/Well-Defined.html). Caranya? Ambil dan dua buah elemen di yang termuat

didalam koset yang sama, dengan kata lain , akan kita tunjukan

Yang b erakibat kesamaan dari

Selanjutnya kita verivikasi merupakan homomorfisma. ambil 2 buah koset dan maka

(Operasi pada )

(definisi dari )

( homomorfisma)

(definisi dari )

Yang terakhir kita buktikan isomorfisma (yaitu bijektive)

Injektif

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1 Votes

Ambil dengan . Itu artinya . Karena homomorfisma

diperoleh berakibat . Ambil maka . Diperoleh

yang berakibat

Surjektif

Ambil sebarang maka terdapat sedemikian hingg a . Karena memetakan ke

diperoleh

QED

Contoh

Diberikan homomorfisma dengan adalah sisa dari pembagian dengan berdasarkan algoritma

pembagian (http://en.wik ipedia.org/wiki/Division_algorithm). Diperoleh kernel dari maka berdasarkan terorema

Isomorfisma Pertama diketahui isomorfik ke

———————————————————————————————————————————————-

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9 thoughts on “Teorema Isomorfisma Pertama”

Related

Lemma Zassenhaus (Lemma

Kupu-kupu)

(https://ariaturns.wordpress.co…

zassenhaus-lemma-kupu-kupu/)

ernel=Subgrup Normal


ormal/)

Pembuktian Teorema

Fundamental Kalkulus


teorema-fundamental-kalkulus/)

SUSANTI DO ALI says:

June 25, 2013 at 7:55 am

bisa bantu saya un tuk teorema isomorfis ketiga beserta buktinya? atas bantuan nya saya ucapkan terima

kasih.

Reply

Aria Turns says:

June 26, 2013 at 7:47 pm

Okey ntar kalo ada waktu (dan mood) saya bahas

Reply

Murniati Suardi says:

May 5, 2013 at 6:19 pm

dibaca ap in G/K? atauu ap mksudx it…

Reply

Aria Turns says:

May 5, 2013 at 8:27 pm

Silahkan klik http://en.wikipedia.org/wiki/Quotient_group#Definition

Reply

hikber says:

https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/comment-page-1/#comment-6274https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/?replytocom=6274#respondhttps://ariaturns.wordpress.com/https://ariaturns.wordpress.com/2012/12/22/pembuktian-teorema-fundamental-kalkulus/https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/comment-page-1/#comment-6129https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/?replytocom=6267#respondhttps://ariaturns.wordpress.com/https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/?replytocom=6129#respondhttp://en.wikipedia.org/wiki/Quotient_group#Definitionhttp://www.facebook.com/mhur.shypoetrybungshuhttps://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/comment-page-1/#comment-6131https://ariaturns.wordpress.com/2010/06/25/kernelsubgrup-normal/https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/?replytocom=6131#respondhttps://ariaturns.wordpress.com/2011/02/10/lemma-zassenhaus-lemma-kupu-kupu/https://ariaturns.wordpress.com/2010/07/09/teorema-isomorfisma-pertama/comment-page-1/#comment-6267


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Silahkan, tinggalkan komentar

December 3, 2012 at 11:38 am

bagaimana dengan jika f suatu homomorfisma dari grup G onto ke grup G’ dengan k ernel I, maka G/I

isomorfik dengan G’ buktinya masih sangat bingunhgg,,,,

Reply

arman says:

August 31, 2012 at 9:51 am

thanks ilmu nya,.. mas, ada refernsi gak untuk pembuktian isomorfisma grup Spesial ortogonal (3) dan

spesial uniter (2).

sblmnya terima kasih, mas

Reply

Aria Turns says:

September 2, 2012 at 4:18 pm

Maaf, tidak ada..

Reply

wiangga0409 says:

July 9, 2010 at 4:09 pm

oh jadi begitu ya, ya saya jadi mengerti sekarang bahwasanya saya benar benar tidak mengerti matematika.

tapi aku senang membaca ini, jadi merasa ikut pintar, gpp kan?

Reply

Aria Turns says:

July 9, 2010 at 8:28 pm

Ini materi kul saya, wajar kalo mbak gak ngerti. Saya hanya menuliskan apa yang yang saya dapat

waktu kuliah, apa itu artinya saya pintar? Gak kan Mbak..

Reply

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