Sylow Theory - Revision history

Kfagerstrom at 01:06, 8 March 2023

2023-03-08T01:06:23Z

Kfagerstrom at 20:38, 17 January 2023

2023-01-17T20:38:05Z

Kfagerstrom: /* Sylow Theorem 1 (Existence) */

2023-01-16T15:50:14Z

Sylow Theorem 1 (Existence)

Kfagerstrom: Created page with " ====Sylow Theorem 1 (Existence) ==== If $p$ is prime and $p^k|n$ , then there exists $H with |H|=p^k "$

2022-12-07T02:37:09Z

Created page with " ====Sylow Theorem 1 (Existence) ==== If <math>p </math> is prime and <math> p^k|n </math>, then there exists <math> H<G </math> with <math>|H|=p^k</math>"

New page

====Sylow Theorem 1 (Existence) ====
If <math>p </math> is prime and <math> p^k|n </math>, then there exists <math> H<G </math> with <math>|H|=p^k</math>

← Older revision		Revision as of 01:06, 8 March 2023
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	<math> \|G\| = p^em </math> where <math>p \not\| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> \|H\| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.		<math> \|G\| = p^em </math> where <math>p \not\| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> \|H\| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
	We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = \| \text{Syl}_p(G)\| </math> to be the number of Sylow p-subgroups.		We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = \| \text{Syl}_p(G)\| </math> to be the number of Sylow p-subgroups.
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		+	[[Category: Group Theory]]

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+:''return to [[817 - Algebra#Sylow Theory| Algebra main page]]
 Let G be a finite group and p a prime. Write the order of <math> G</math> as
 <math> |G| = p^em </math> where <math>p \not| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
 We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups.

← Older revision		Revision as of 15:50, 16 January 2023
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−	~~====Sylow Theorem 1 (Existence) ====~~	+	Let G be a finite group and p a prime. Write the order of <math> G</math> as
−	If <math>p </math> ~~is prime and~~ <math> p^k\|n </math>~~, then there exists~~ <math> H<G </math> ~~with~~ <math>\|H\|=p^k</math>	+	<math> \|G\| = p^em </math> where <math>p \not\| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> \|H\| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
		+	We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = \| \text{Syl}_p(G)\| </math> to be the number of Sylow p-subgroups.

Sylow Theory - Revision history

Kfagerstrom at 01:06, 8 March 2023

Kfagerstrom at 20:38, 17 January 2023

Kfagerstrom: /* Sylow Theorem 1 (Existence) */

Kfagerstrom: Created page with " ====Sylow Theorem 1 (Existence) ==== If p is prime and p^k|n , then there exists H with |H|=p^k"

Kfagerstrom: Created page with " ====Sylow Theorem 1 (Existence) ==== If $p$ is prime and $p^k|n$ , then there exists $H with |H|=p^k "$