List questions from sylow-theory

Published on 15 Apr 2026 - 15:48

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Putnam 2007 A5: Finite group $n$ elements order $p$, prove either $n=0$ or $p$ divides $n+1$

Published on 16 Apr 2026 - 16:20

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Find two finite groups $G_1$ and $G_2$ such as:

Published on 15 Apr 2026 - 23:34

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Let $G=\langle (1,2,3,4,5,6,7), (2,4,3,7,5,6) \rangle$

Published on 16 Apr 2026 - 11:19

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Let $A \unlhd G$, then $n_p(G)=n_p(G/A)n_p(PA)$ for $P\in Syl_p(G)$ and $n_p(PA)=|A:N_A(P)|$

Published on 16 Apr 2026 - 17:58

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$3$-Sylow subgroup of $SL(n, \mathbb{F}_3)$

Published on 17 Apr 2026 - 0:05

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A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple.

Published on 10 May 2026 - 13:26

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Finite group with cyclic Sylow-2 group has a normal subgroup of index 2.

Published on 14 Apr 2026 - 11:11

667

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Understanding Proof of Sylow theorem from Herstein

Published on 08 Apr 2026 - 18:19

99

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First Sylow Theorem proof

Published on 11 Apr 2026 - 16:21

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Can one prove that a group of order 150 is not simple using element counting?

Published on 10 May 2026 - 13:25

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Why does $A_5$ have $\binom{5}{4}$ Sylow 2-subgroups?

Published on 14 Apr 2026 - 22:25

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Every Abelian group with order a product of two different primes is cyclic.

Published on 11 Apr 2026 - 17:38

63

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How to guarantee non existence of order 4 elements in class group of a maximal order

Published on 07 May 2026 - 9:19

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A group of order $36$ has either a normal Sylow $2$-subgroup or a normal Sylow $3$-subgroup

Published on 15 Apr 2026 - 5:52

List Question

P-sylow subgroups of group of order $340$

Putnam 2007 A5: Finite group $n$ elements order $p$, prove either $n=0$ or $p$ divides $n+1$

Find two finite groups $G_1$ and $G_2$ such as:

Let $G=\langle (1,2,3,4,5,6,7), (2,4,3,7,5,6) \rangle$

Let $A \unlhd G$, then $n_p(G)=n_p(G/A)n_p(PA)$ for $P\in Syl_p(G)$ and $n_p(PA)=|A:N_A(P)|$

$3$-Sylow subgroup of $SL(n, \mathbb{F}_3)$

A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple.

Finite group with cyclic Sylow-2 group has a normal subgroup of index 2.

Understanding Proof of Sylow theorem from Herstein

First Sylow Theorem proof

Can one prove that a group of order 150 is not simple using element counting?

Why does $A_5$ have $\binom{5}{4}$ Sylow 2-subgroups?

Every Abelian group with order a product of two different primes is cyclic.

How to guarantee non existence of order 4 elements in class group of a maximal order

A group of order $36$ has either a normal Sylow $2$-subgroup or a normal Sylow $3$-subgroup

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