Question

Let $G$ be a group of order $30$ and $A, B$ be two normal subgroups of $G$ of order $2$ and $5$ respectively. Show that $G/AB$ contains $3$ elements.

score 432 · Answer 1 · 2026-03-29 15:33:20

432

Views

Let $G$ be a group of order $30$ and $A, B$ be two normal subgroups of $G$ of order $2$ and $5$ respectively. Show that $G/AB$ contains $3$ elements.

Published on 29 Mar 2026 - 15:33

score 123 · Answer 2 · 2022-07-05 23:04:28

123

Views

Prove that any automorphism $\phi \in Aut(\mathbb{Z}_n)$ is determined by $\phi([1])$ and that $\phi([1])$ must be a generator for $\mathbb{Z}_n$

Published on 05 Jul 2022 - 23:04

score 204 · Answer 3 · 2022-07-07 06:31:16

204

Views

Why is $\{1,-1,i,-i\}$ isomorphic to the cyclic group $C_4$?

Published on 07 Jul 2022 - 6:31

score 92 · Answer 4 · 2022-07-10 09:12:36

92

Views

Cyclic groups and abelian groups

Published on 10 Jul 2022 - 9:12

score 131 · Answer 5 · 2022-07-10 11:02:45

131

Views

Showing $G$ is cyclic if $G$ has two proper subgroup

Published on 10 Jul 2022 - 11:02

score 80 · Answer 6 · 2022-07-15 00:17:03

80

Views

Why finite subgroups of $\Bbb C^*$ are cyclic?

Published on 15 Jul 2022 - 0:17

score 69 · Answer 7 · 2022-07-21 15:22:17

69

Views

Suppose that $G$ is a finite cyclic group and $|G|=n$. If $d \mid n$, then the number of elements of order $d$ in $G$ is $\varphi(d)$.

Published on 21 Jul 2022 - 15:22

score 68 · Answer 8 · 2022-07-27 22:18:52

68

Views

Let a cyclic group $A=\langle a\rangle$ act on a group $G$ whose order is odd. If $[G,a^2]=1$, then $\{x\in G: x=x^{-a}\}=\{[x,a]:x\in G\}.$

Published on 27 Jul 2022 - 22:18

score 161 · Answer 9 · 2026-03-28 09:33:40

161

Views

Compute the number of distinct actions of cyclic group $C_n$ on a set $X,$ s.t $|X|= n+1.$

Published on 28 Mar 2026 - 9:33

score 231 · Answer 10 · 2022-08-05 06:59:30

Let $G$ be a group, $|G|=n$. Suppose $\forall d\in \mathbb{N}$ such that $d\mid n$, there are at most $d$ elements s.t. $x^d=1$. Prove $G$ is cyclic.

score 131 · Answer 11 · 2026-03-28 09:33:43

131

Views

Does a finite-by-(infinite dihedral) group have the form $N\rtimes H$ for a finite normal subgroup $N$ and a cyclic subgroup $H$?

Published on 28 Mar 2026 - 9:33

score 69 · Answer 12 · 2026-03-30 03:55:13

69

Views

How to define given conditions related to generators of groups on GAP

Published on 30 Mar 2026 - 3:55

score 97 · Answer 13 · 2022-08-11 09:06:11

97

Views

Inverse of $3$ in multiplicative group $C_{20}.$

Published on 11 Aug 2022 - 9:06

score 40 · Answer 14 · 2022-08-15 19:22:07

40

Views

What are the elements forming the subgroup $H = \langle a,b \rangle$ of $\Bbb C^\ast$ when $a = e^{2\pi i/5}$ and $b=e^{2\pi i/7}$? Is $H$ cyclic?

Published on 15 Aug 2022 - 19:22

score 66 · Answer 15 · 2022-08-21 08:14:21

66

Views

Let $a = 2/3$ and $b=5/7$ and let $G = \langle a,b \rangle \subset (\Bbb Q, +)$. Show that $G$ is cyclic and generated by $1/21$.

Published on 21 Aug 2022 - 8:14

List Question

Let $G$ be a group of order $30$ and $A, B$ be two normal subgroups of $G$ of order $2$ and $5$ respectively. Show that $G/AB$ contains $3$ elements.

Prove that any automorphism $\phi \in Aut(\mathbb{Z}_n)$ is determined by $\phi([1])$ and that $\phi([1])$ must be a generator for $\mathbb{Z}_n$

Why is $\{1,-1,i,-i\}$ isomorphic to the cyclic group $C_4$?

Cyclic groups and abelian groups

Showing $G$ is cyclic if $G$ has two proper subgroup

Why finite subgroups of $\Bbb C^*$ are cyclic?

Suppose that $G$ is a finite cyclic group and $|G|=n$. If $d \mid n$, then the number of elements of order $d$ in $G$ is $\varphi(d)$.

Let a cyclic group $A=\langle a\rangle$ act on a group $G$ whose order is odd. If $[G,a^2]=1$, then $\{x\in G: x=x^{-a}\}=\{[x,a]:x\in G\}.$

Compute the number of distinct actions of cyclic group $C_n$ on a set $X,$ s.t $|X|= n+1.$

Let $G$ be a group, $|G|=n$. Suppose $\forall d\in \mathbb{N}$ such that $d\mid n$, there are at most $d$ elements s.t. $x^d=1$. Prove $G$ is cyclic.

Does a finite-by-(infinite dihedral) group have the form $N\rtimes H$ for a finite normal subgroup $N$ and a cyclic subgroup $H$?

How to define given conditions related to generators of groups on GAP

Inverse of $3$ in multiplicative group $C_{20}.$

What are the elements forming the subgroup $H = \langle a,b \rangle$ of $\Bbb C^\ast$ when $a = e^{2\pi i/5}$ and $b=e^{2\pi i/7}$? Is $H$ cyclic?

Let $a = 2/3$ and $b=5/7$ and let $G = \langle a,b \rangle \subset (\Bbb Q, +)$. Show that $G$ is cyclic and generated by $1/21$.

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