Question

Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.

score 346 · Answer 1 · 2026-04-15 04:50:35

346

Views

Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.

Published on 15 Apr 2026 - 4:50

score 234 · Answer 2 · 2026-05-10 13:08:58

234

Views

$G$ is a group of order $pq$ and $P_q$ and $P_p$ are Sylow subgroups...

Published on 10 May 2026 - 13:08

score 66 · Answer 3 · 2026-04-14 05:50:56

66

Views

Show that $T$ is normal in $GL_{2}(\Bbb{R})$.

Published on 14 Apr 2026 - 5:50

score 315 · Answer 4 · 2026-04-13 17:30:18

315

Views

Character table only with knowledge of (normal) subgroups

Published on 13 Apr 2026 - 17:30

score 177 · Answer 5 · 2026-04-13 08:11:59

177

Views

Is a subnormal Hall subgroup of a finite group normal?

Published on 13 Apr 2026 - 8:11

score 1.4k · Answer 6 · 2026-04-11 23:11:00

1.4k

Views

Applications of the strong Cayley theorem

Published on 11 Apr 2026 - 23:11

score 32 · Answer 7 · 2026-05-10 13:24:35

32

Views

A normal minimal subgroup in a Sylow 2-subgroup

Published on 10 May 2026 - 13:24

score 110 · Answer 8 · 2026-04-14 20:53:24

110

Views

Excercise 5.D.4 from M.Isaacs Finite Group Theory and Tate's Theorem

Published on 14 Apr 2026 - 20:53

score 1.5k · Answer 9 · 2026-04-14 16:26:50

1.5k

Views

Prove that the group of diagonal matrices is not a normal subgroup of the group $\mathrm{GL}_n\,(\mathbb K)$ if $n\ge 2$ and $|K|\ge 3$

Published on 14 Apr 2026 - 16:26

score 156 · Answer 10 · 2026-05-07 06:54:17

156

Views

Is the intersection of two subgroups, defined below, always trivial?

Published on 07 May 2026 - 6:54

score 895 · Answer 11 · 2026-04-16 11:19:13

895

Views

Let $H$ be a subgroup of a group $G$. Then prove that $Ha$ or $aH$ is a subgroup of $G$ if and only if $a\in H.$

Published on 16 Apr 2026 - 11:19

score 51 · Answer 12 · 2026-04-08 11:15:17

51

Views

How do I prove that the subset of unfixed transpositions from $A_4$ is normal to $A_4$?

Published on 08 Apr 2026 - 11:15

score 709 · Answer 13 · 2026-04-18 11:22:09

709

Views

Infinite case: Let $N$ be a normal subgroup of index m in $G$. Prove that $a^{m}\in N$ for all $a\in G$

Published on 18 Apr 2026 - 11:22

score 1.3k · Answer 14 · 2026-04-16 06:30:17

1.3k

Views

Let M and N be normal subgroups of a group G such that $M \cap N=\{e\}$. Prove that $mn=nm$ for all $m\in M$ and all $n\in N$.

Published on 16 Apr 2026 - 6:30

score 1.1k · Answer 15 · 2026-04-16 16:58:58

1.1k

Views

Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Prove that $x^2 \in Z(G)$

Published on 16 Apr 2026 - 16:58

List Question

Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.

$G$ is a group of order $pq$ and $P_q$ and $P_p$ are Sylow subgroups...

Show that $T$ is normal in $GL_{2}(\Bbb{R})$.

Character table only with knowledge of (normal) subgroups

Is a subnormal Hall subgroup of a finite group normal?

Applications of the strong Cayley theorem

A normal minimal subgroup in a Sylow 2-subgroup

Excercise 5.D.4 from M.Isaacs Finite Group Theory and Tate's Theorem

Prove that the group of diagonal matrices is not a normal subgroup of the group $\mathrm{GL}_n\,(\mathbb K)$ if $n\ge 2$ and $|K|\ge 3$

Is the intersection of two subgroups, defined below, always trivial?

Let $H$ be a subgroup of a group $G$. Then prove that $Ha$ or $aH$ is a subgroup of $G$ if and only if $a\in H.$

How do I prove that the subset of unfixed transpositions from $A_4$ is normal to $A_4$?

Infinite case: Let $N$ be a normal subgroup of index m in $G$. Prove that $a^{m}\in N$ for all $a\in G$

Let M and N be normal subgroups of a group G such that $M \cap N=\{e\}$. Prove that $mn=nm$ for all $m\in M$ and all $n\in N$.

Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Prove that $x^2 \in Z(G)$

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