Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show if $p$ does not divide $[G:H]$, then $P \subseteq H$

score 65 · Answer 6 · 2015-03-23 12:04:13

65

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Why is the group abelian?

Published on 23 Mar 2015 - 12:04

score 837 · Answer 7 · 2015-03-23 15:04:14

837

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Find the number of Sylow $p$-subgroups of $G$, if we know that $\lvert G\rvert=6$

Published on 23 Mar 2015 - 15:04

score 422 · Answer 8 · 2015-03-25 17:09:07

422

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Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Published on 25 Mar 2015 - 17:09

score 358 · Answer 9 · 2015-03-30 18:27:56

358

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Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

Published on 30 Mar 2015 - 18:27

score 582 · Answer 10 · 2015-03-31 23:48:56

582

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The number of Sylow subgroups on $G$ with $|G|=pqr$

Published on 31 Mar 2015 - 23:48

score 617 · Answer 11 · 2015-04-01 01:14:29

617

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Sylow $p$-group of a transitive subgroup of $S_p$

Published on 01 Apr 2015 - 1:14

score 245 · Answer 12 · 2015-04-01 01:42:09

245

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Sylow $p$-subgroup of a normal group.

Published on 01 Apr 2015 - 1:42

score 172 · Answer 13 · 2015-04-01 20:23:57

172

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Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

Published on 01 Apr 2015 - 20:23

score 354 · Answer 14 · 2015-04-05 20:55:58

354

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If we have exactly $8$ Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Published on 05 Apr 2015 - 20:55

score 2.5k · Answer 15 · 2015-04-06 10:06:21

2.5k

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Why does group action by conjugation on sylow subgroups define a homomorphism into the symmetric group?

Published on 06 Apr 2015 - 10:06

List Question

Exercise 257 from Rose" A course on group theory"

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$

Let G be a group of order $p^2q^2$ where $p > q$ are primes. If $|G| \ne 36$, show that G has exactly one Sylow p-subgroup

About first Sylow Theorem proof

Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show if $p$ does not divide $[G:H]$, then $P \subseteq H$

Why is the group abelian?

Find the number of Sylow $p$-subgroups of $G$, if we know that $\lvert G\rvert=6$

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

The number of Sylow subgroups on $G$ with $|G|=pqr$

Sylow $p$-group of a transitive subgroup of $S_p$

Sylow $p$-subgroup of a normal group.

Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

If we have exactly $8$ Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Why does group action by conjugation on sylow subgroups define a homomorphism into the symmetric group?

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