2020-2ํ•™๊ธฐ, ๋Œ€ํ•™์—์„œ โ€˜ํ˜„๋Œ€๋Œ€์ˆ˜1โ€™ ์ˆ˜์—…์„ ๋“ฃ๊ณ  ๊ณต๋ถ€ํ•œ ๋ฐ”๋ฅผ ์ •๋ฆฌํ•œ ๊ธ€์ž…๋‹ˆ๋‹ค. ์ง€์ ์€ ์–ธ์ œ๋‚˜ ํ™˜์˜์ž…๋‹ˆ๋‹ค :)

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2020-2ํ•™๊ธฐ, ๋Œ€ํ•™์—์„œ โ€˜ํ˜„๋Œ€๋Œ€์ˆ˜1โ€™ ์ˆ˜์—…์„ ๋“ฃ๊ณ  ๊ณต๋ถ€ํ•œ ๋ฐ”๋ฅผ ์ •๋ฆฌํ•œ ๊ธ€์ž…๋‹ˆ๋‹ค. ์ง€์ ์€ ์–ธ์ œ๋‚˜ ํ™˜์˜์ž…๋‹ˆ๋‹ค :)



3๊ฐ€์ง€ Isomorphism Threorem์ด ์ˆœํ•œ ๋ง›์ด์—ˆ๋‹ค๋ฉด, 3๊ฐ€์ง€ Sylow Theorem์€ ๋งค์šด ๋ง›์ด๋‹ค ใ… ใ… 

๊ทธ๋‚˜๋งˆ ๋‹คํ–‰์ธ ์ ์€ Sylow Theorem์— ๋Œ€ํ•œ ์ฆ๋ช…์€ ํ•™๋ถ€ ์ˆ˜์ค€์„ ๋ฒ—์–ด๋‚˜์„œ ์ˆ˜์—… ๋•Œ ๊ต์ˆ˜๋‹˜์ด ์ฆ๋ช…ํ•˜์‹œ์ง„ ์•Š์•˜๋‹ค๋Š” ์ ์ด๋‹ค ใ… ใ… 

๊ทธ๋ž˜๋„ Application์€ ์—ฌ์ „ํžˆ ์–ด๋ ต๋‹ค ใ… ใ… 


Sylow Theorem๋“ค์€ ๋ชจ๋‘ ์œ ํ•œ๊ตฐ์„ ๋ถ„์„ํ•˜๋Š” ๋„๊ตฌ๋กœ ์‚ฌ์šฉ๋œ๋‹ค. ๊ทธ๋ฆฌ๊ณ  ์†Œ์ˆ˜ p๋ฅผ ์•„์ฃผ ์ข‹์•„ํ•œ๋‹ค

Sylow Theorem์„ ์ด์šฉํ•˜๋ฉด ์œ ํ•œ๊ตฐ์—์„œ ์œ„์ˆ˜์— ๋Œ€ํ•œ ์ •๋ณด ๋งŒ์œผ๋กœ ์œ ํ•œ๊ตฐ์˜ ๊ตฌ์กฐ์™€ ๋ถ€๋ถ„๊ตฐ๋“ค์— ๋Œ€ํ•ด ์•Œ์•„๋‚ผ ์ˆ˜ ์žˆ๋‹ค!!

Lagrange Theorem์ด ํŠน์ • ์œ„์ˆ˜์˜ ๋ถ€๋ถ„๊ตฐ์ด ์—†์Œ์„ ๋ณด์ด๋Š” ๋ฐ์— ์œ ์šฉํ–ˆ๋‹ค๋ฉด,
(Converse of Lagrange ๋•Œ๋ฌธ์— Lagrange Thm์œผ๋กœ ํŠน์ • ์œ„์ˆ˜์˜ ๋ถ€๋ถ„๊ตฐ์ด ์žˆ์Œ์„ ๋ณด์ด๋Š” ๊ฑด ๋ถˆ๊ฐ€๋Šฅํ•˜๋‹ค.)

Sylow Theorem์€ ํŠน์ • ์œ„์ˆ˜์˜ ๋ถ€๋ถ„๊ตฐ์ด ์žˆ์Œ์„ ๋ณด์ด๋Š” ๋ฐ์— ์œ ์šฉํ•˜๋‹ค!!



p-groupPermalink

Definition. p-group

Let p be a prime number.

A group G is called โ€œp-groupโ€,

if every elts has a power of p order

p.s. p-group์˜ subgroup๋„ ์—ฌ์ „ํžˆ p-group์ด๋‹ค!
(์ƒ๊ฐํ•ด๋ณด๋ฉด, p-group์˜ ์›์†Œ๋ฅผ ๋ชจ์•„ subgroup์„ ๋งŒ๋“ค์—ˆ์œผ๋‹ˆ ๋‹น์—ฐํ•˜๊ธด ํ•˜๋‹ค ใ…‹ใ…‹)



Theorem. (Cauchy)

Let G be a finite group, and pโˆฃ|G|.

Then, G has an elt of order p.

์ˆ˜์—… ๋•Œ ์ฆ๋ช…์€ ์ƒ-๋žตํ•˜์…จ์Œ!


Corollary.

Let G be a finite group,

G is a p-group โŸบ |G| is a power of p


(โŸธ) Let |G|=pn.

For aโˆˆG, |a|โˆฃ|G|,

then |a|=pk(0โ‰คkโ‰คn).

This means G is a p-group.

(โŸน) Supp. G is p-group.

Then, p is the only prime divisor of |G|.

If thereโ€™s additional prime divisor, pโ€ฒโ‰ p,

then by Cauchy, โˆƒ an elt aโˆˆG s.t. |a|=pโ€ฒ.

then this mean G is not p-group.

Therefore, p is the only prime divisor of |G|.

This means |G|=pn.



NormalizerPermalink

Definition.

For Hโ‰คG, the normalizer of H in G is

NG(H):={gโˆˆGโˆฃgHgโˆ’1=H}

๋งŒ์•ฝ NG(H)=G๋ผ๋ฉด, subgroup H๋Š” normal subgroup์ด ๋œ๋‹ค!


Normalizer NG(H)์— ๋Œ€ํ•œ ์„ฑ์งˆ๋“ค์„ ์ข€๋” ์•Œ์•„๋ณด์ž.

Properties. Normalizer NG(H)

1. HโŠดNG(H)

normalizer ์ •์˜๊ฐ€ H๋ฅผ normalํ•˜๊ฒŒ ๋งŒ๋“œ๋Š” ์›์†Œ๋“ค์˜ ์ง‘ํ•ฉ์ด๋ฏ€๋กœ ๋‹น์—ฐํžˆ H๋Š” NG(H)์— normal subgroup์ด๋‹ค.


2. NG(H)โ‰คG

NG(H)๊ฐ€ subgroup์˜ ์ •์˜๋ฅผ ๋งŒ์กฑํ•˜๋Š”์ง€ ํ™•์ธํ•ด๋ณด๋ฉด ๋œ๋‹ค.



1st Sylow TheoremPermalink

Theorem. 1st Sylow Theorem

Let |G|=pnโ‹…m where n,mโˆˆN and (m,p)=1.

Then,

1. G contains subgroup of order pi(0โ‰คiโ‰คn).

  • p0=e : trivial subgroup
  • p1 : by Cauchy
  • p2,โ‹ฏpn : by 1st Sylow Thm


2. Every โ€œsubgroup of order pi(i<n)โ€ is a normal subgroup of a โ€œsubgroup of order pi+1โ€.

์ฆ‰, |H|=pi์ธ subgp H์— ๋Œ€ํ•ด HโŠดHโ€ฒ์ธ normal subgroup์ด |Hโ€ฒ|=pi+1=|H|โ‹…p๋ฅผ ๊ฐ–๊ณ  ์กด์žฌํ•จ์„ ๋ณด์žฅํ•œ๋‹ค.

์ฆ๋ช…์€ ํ•™๋ถ€ ๊ณผ์ •์„ ์ƒํšŒํ•˜๋ฏ€๋กœ ์ƒ-๋žต ํ•œ๋‹ค.



Sylow p-groupPermalink

Definition. Sylow p-group

For |G|<โˆž,

a โ€œSylow p-group Pโ€ is a maximal p-subgroup of G.

์ฆ‰, p-subgroup์ธ๋ฐ ์ž๊ธฐ ์ž์‹ ์„ ํฌํ•จํ•˜๋Š” ๋” ํฐ p-subgroup์ด ์—†๋Š” p-subgroup์„ maximal subgroup์ด๋ผ๊ณ  ํ•˜๋ฉฐ, ์ด๊ฒƒ์„ โ€œSylow p-groupโ€์ด๋ผ๊ณ  ํ•œ๋‹ค.

maximal p-subgroup = Sylow p-group

โ€ป ์–ด๋–ค Group์ด๋“  Sylow p-subgroup์„ ๋ฐ˜๋“œ์‹œ ๊ฐ€์ง„๋‹ค!



2nd Sylow TheoremPermalink

Theorem. 2nd Sylow Theorem

Let P1, P2 be two Sylow p-subgroup of finite group G.

Then, P1 and P2 are conjugate of each other.

์ฆ‰,

P1=gP2gโˆ’1(for somegโˆˆG)



3rd Sylow TheoremPermalink

Theorem. 3rd Sylow Theorem

If a prime p divides |G|; pโˆฃ|G|,

then the (# of Sylow p-subgroup) is congruent to 1 (mod p)

and it divides |G|.




Sylow Theorem์— ๋Œ€ํ•œ Application์€ ์•„๋ž˜์—์„œ ํ™•์ธํ•  ์ˆ˜ ์žˆ๋‹ค!