List questions from normal-subgroups

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Related to L1 norm condition for the product of a stochastic vector and a stochastic matrix

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$(\mathbb{F}_p^\times)^2$ subgroup of $\mathbb{F}_p^\times$ and $(\mathbb{F}_p^\times)/(\mathbb{F}_p^\times)^2 \cong \mathbb{Z}/ 2\mathbb{Z}$

Published on 09 Mar 2020 - 18:49

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Theorem 3.19 (The Third Isomorphism Theorem) from Dummit's Abstract Algebra

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A question about Lang's proof of simplicity of $A_n$

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Asking the proof of maximal normal subgroup in this answer: https://math.stackexchange.com/a/161593/435467

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$|G|=p^n m$ , $p$ is prime and $\gcd(p,m)=1$. Let $H$ normal in $G$ of order $p^n$ and $K$ is a subgroup of $G$ of order $p^k$ then $K\subseteq H$

Published on 16 Mar 2020 - 16:28

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Prove $\varphi (m,n) = (mN, nM)$ is injective

Published on 17 Mar 2020 - 15:36

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A finite product of nilpotent groups is nilpotent

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Series of nilpotent group and one of its subgroups

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$|G|<\infty$. $p$ the smallest prime dividing $|G|$. $|G:H|=p$. then H is normal.

Published on 23 Mar 2020 - 9:45

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Decreasing sequence of normal subgroups of a group $G$

Published on 26 Mar 2026 - 1:10

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$G$ has a unique normal subgroup of order $p$ iff $G$ is cyclic center.

Published on 25 Mar 2026 - 20:53

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Normal subgroups of k-transitive groups.

Published on 26 Mar 2020 - 1:02

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Normal subgroups and Normalizers

Published on 26 Mar 2020 - 13:05

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Exercises about subgroup generated by $[h,k]$

Related to L1 norm condition for the product of a stochastic vector and a stochastic matrix

$(\mathbb{F}_p^\times)^2$ subgroup of $\mathbb{F}_p^\times$ and $(\mathbb{F}_p^\times)/(\mathbb{F}_p^\times)^2 \cong \mathbb{Z}/ 2\mathbb{Z}$

Theorem 3.19 (The Third Isomorphism Theorem) from Dummit's Abstract Algebra

A question about Lang's proof of simplicity of $A_n$

Asking the proof of maximal normal subgroup in this answer: https://math.stackexchange.com/a/161593/435467

$|G|=p^n m$ , $p$ is prime and $\gcd(p,m)=1$. Let $H$ normal in $G$ of order $p^n$ and $K$ is a subgroup of $G$ of order $p^k$ then $K\subseteq H$

Prove $\varphi (m,n) = (mN, nM)$ is injective

A finite product of nilpotent groups is nilpotent

Series of nilpotent group and one of its subgroups

$|G|<\infty$. $p$ the smallest prime dividing $|G|$. $|G:H|=p$. then H is normal.

Decreasing sequence of normal subgroups of a group $G$

$G$ has a unique normal subgroup of order $p$ iff $G$ is cyclic center.

Normal subgroups of k-transitive groups.

Normal subgroups and Normalizers

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