List questions from characteristic-subgroups

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Quick way to determine characteristic subgroups

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Fully invariant subgroups of the group $S_4$

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Verbal Subgroups of $\mathbb{Z}^2$

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Let $G$ be f.g. with $H\le G$ s.t. $[G:H]<\infty$. Then $\exists K\le H$ with $K\sqsubseteq G$ and $[G:K]<\infty$.

Published on 23 Feb 2026 - 7:24

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Lemma of Characterictic Subgroups

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$\{0\}\times 3\mathbb{Z}/9\mathbb{Z}$ as a characteristic subgroup of $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z}$

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The characteristic subgroup of order $m$ (odd) in a group of order $2^nm$ with cyclic 2-sylow

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Normal subgroup of a characteristic subgroup

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If a subgroup is unique, is it characteristic?

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If $G=\langle H,K\rangle$, then $G'=[H,K]$

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If $H\trianglelefteq K$ and $K\text{ char } G$, then is $H\trianglelefteq G$?

Published on 23 Feb 2026 - 7:21

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Given $N$ char $G$, prove that $N\lt K \lt G$ and $K/N$ char $G/N$ implies $K$ char $G$

Published on 23 Feb 2026 - 7:23

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Conjecture: if $H$ is characteristic in $G$ and $K\leq G$ such that $K\cong H$, then $H=K$.

Published on 23 Feb 2026 - 7:25

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Isomorphism on a Sylow-$p$-subgroup of a group $G$.

Published on 23 Feb 2026 - 7:21

List Question

A simple question on characteristic subgroups

Quick way to determine characteristic subgroups

Fully invariant subgroups of the group $S_4$

Verbal Subgroups of $\mathbb{Z}^2$

Let $G$ be f.g. with $H\le G$ s.t. $[G:H]<\infty$. Then $\exists K\le H$ with $K\sqsubseteq G$ and $[G:K]<\infty$.

Lemma of Characterictic Subgroups

$\{0\}\times 3\mathbb{Z}/9\mathbb{Z}$ as a characteristic subgroup of $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z}$

The characteristic subgroup of order $m$ (odd) in a group of order $2^nm$ with cyclic 2-sylow

Normal subgroup of a characteristic subgroup

If a subgroup is unique, is it characteristic?

If $G=\langle H,K\rangle$, then $G'=[H,K]$

If $H\trianglelefteq K$ and $K\text{ char } G$, then is $H\trianglelefteq G$?

Given $N$ char $G$, prove that $N\lt K \lt G$ and $K/N$ char $G/N$ implies $K$ char $G$

Conjecture: if $H$ is characteristic in $G$ and $K\leq G$ such that $K\cong H$, then $H=K$.

Isomorphism on a Sylow-$p$-subgroup of a group $G$.

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