List questions from p-groups

Published on 04 Nov 2021 - 22:22

134

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Examples of non abelian powerful p-groups

Published on 09 Nov 2021 - 14:50

281

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Understanding the normalizer of a Sylow $p$-subgroup

Published on 27 Nov 2021 - 5:36

91

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A group acting on a $p$-group

Published on 29 Nov 2021 - 17:37

126

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Question about a proof of: If $P$ is a finite $p$-group and $\langle 1\rangle\neq N\trianglelefteq P$, then $N\cap Z(P)\neq\langle 1\rangle$.

Published on 07 Dec 2021 - 0:37

58

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The centre of a minimal nonabelian $p$-subgroup

Published on 13 Dec 2021 - 12:59

110

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Prove that there exists a $z \in Z(G)$ such that $yx = zxy$.

Published on 16 Dec 2021 - 19:58

76

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On p-group of maximal class

Published on 17 Dec 2021 - 1:00

599

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Let $G$ be a finite $p$-group, where $p$ is prime. Show that if $G$ is not cyclic, then $G$ has at least $p+1$ maximal subgroups.

Published on 25 Mar 2026 - 15:42

166

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There exist infinite solvable $p$-groups with trivial centre. (Use a hint.)

Published on 25 Mar 2026 - 14:31

133

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For a group $G$ of order $p^n$, $G\cong H$ for some $H\le\Bbb Z_p\wr\dots\wr\Bbb Z_p$.

Published on 25 Mar 2026 - 14:35

237

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Find the upper central series of $Q_{2^n}$.

Published on 31 Dec 2021 - 16:44

215

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How many cyclic subgroups are in $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$?

Published on 01 Jan 2022 - 2:04

173

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Proving that if $|G|=p^n$ then $\exists a \in G:|C(a)| = p^{n-1}$

Published on 05 Jan 2022 - 11:08

231

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Understanding the definition of Sylow $p$-subgroups

Published on 25 Mar 2026 - 17:26

List Question

Prove that $|Z(G)|=p$

Examples of non abelian powerful p-groups

Understanding the normalizer of a Sylow $p$-subgroup

A group acting on a $p$-group

Question about a proof of: If $P$ is a finite $p$-group and $\langle 1\rangle\neq N\trianglelefteq P$, then $N\cap Z(P)\neq\langle 1\rangle$.

The centre of a minimal nonabelian $p$-subgroup

Prove that there exists a $z \in Z(G)$ such that $yx = zxy$.

On p-group of maximal class

Let $G$ be a finite $p$-group, where $p$ is prime. Show that if $G$ is not cyclic, then $G$ has at least $p+1$ maximal subgroups.

There exist infinite solvable $p$-groups with trivial centre. (Use a hint.)

For a group $G$ of order $p^n$, $G\cong H$ for some $H\le\Bbb Z_p\wr\dots\wr\Bbb Z_p$.

Find the upper central series of $Q_{2^n}$.

How many cyclic subgroups are in $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$?

Proving that if $|G|=p^n$ then $\exists a \in G:|C(a)| = p^{n-1}$

Understanding the definition of Sylow $p$-subgroups

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