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 343 · Answer 1 · 2018-03-21 15:15:34

343

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 21 Mar 2018 - 15:15

score 223 · Answer 2 · 2026-03-22 18:25:45

223

Views

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

Published on 22 Mar 2026 - 18:25

score 63 · Answer 3 · 2018-03-21 19:49:09

63

Views

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

Published on 21 Mar 2018 - 19:49

score 313 · Answer 4 · 2026-03-25 04:37:46

313

Views

Character table only with knowledge of (normal) subgroups

Published on 25 Mar 2026 - 4:37

score 174 · Answer 5 · 2018-03-29 12:07:20

174

Views

Is a subnormal Hall subgroup of a finite group normal?

Published on 29 Mar 2018 - 12:07

score 1.4k · Answer 6 · 2026-03-25 04:39:17

1.4k

Views

Applications of the strong Cayley theorem

Published on 25 Mar 2026 - 4:39

score 29 · Answer 7 · 2026-03-24 23:44:34

29

Views

A normal minimal subgroup in a Sylow 2-subgroup

Published on 24 Mar 2026 - 23:44

score 108 · Answer 8 · 2026-03-25 01:17:19

108

Views

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

Published on 25 Mar 2026 - 1:17

score 1.5k · Answer 9 · 2018-04-08 08:31:57

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 08 Apr 2018 - 8:31

score 150 · Answer 10 · 2026-03-22 13:46:54

150

Views

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

Published on 22 Mar 2026 - 13:46

score 892 · Answer 11 · 2018-04-15 19:17:49

892

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 15 Apr 2018 - 19:17

score 49 · Answer 12 · 2018-04-17 02:08:58

49

Views

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

Published on 17 Apr 2018 - 2:08

score 706 · Answer 13 · 2026-03-20 04:51:40

706

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 20 Mar 2026 - 4:51

score 1.2k · Answer 14 · 2018-04-18 14:41:03

1.2k

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 18 Apr 2018 - 14:41

score 1.1k · Answer 15 · 2018-04-18 16:49:02

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 18 Apr 2018 - 16:49

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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