Question

Given a finite group $G$ and $N$ a normal subgroup such that $N\cong K$ where $K$ is Klein group and $G/N\cong \Bbb{Z}/7\Bbb{Z}$, prove $N\leq Z(G)$

score 48 · Answer 1 · 2022-06-27 15:54:42

48

Views

Given a finite group $G$ and $N$ a normal subgroup such that $N\cong K$ where $K$ is Klein group and $G/N\cong \Bbb{Z}/7\Bbb{Z}$, prove $N\leq Z(G)$

Published on 27 Jun 2022 - 15:54

score 82 · Answer 2 · 2022-07-01 10:56:46

82

Views

Show that $\Bbb Z_{12}/\langle[3]_{12}\rangle \cong \Bbb Z_p$ for some $p \in \Bbb N$.

Published on 01 Jul 2022 - 10:56

score 431 · Answer 3 · 2022-07-02 13:39:47

431

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 02 Jul 2022 - 13:39

score 61 · Answer 4 · 2022-07-04 16:57:42

61

Views

Finite order elements of quotient group ${G}/{G^m}$

Published on 04 Jul 2022 - 16:57

score 58 · Answer 5 · 2022-07-05 07:41:14

58

Views

Extension of a homomorphism

Published on 05 Jul 2022 - 7:41

score 68 · Answer 6 · 2022-07-06 08:50:20

68

Views

Factor group of a factor groups

Published on 06 Jul 2022 - 8:50

score 262 · Answer 7 · 2022-07-27 20:20:00

262

Views

On a proof that the quaternion $8$-group $Q$ mod its center is isomorphic to the Klein four group

Published on 27 Jul 2022 - 20:20

score 391 · Answer 8 · 2022-07-29 19:34:18

391

Views

Prove that $A_5$ does not have a subgroup of order 15, 20, nor 30 (without using simple group and Sylow's Theorem)

Published on 29 Jul 2022 - 19:34

score 149 · Answer 9 · 2026-03-28 14:55:32

149

Views

Smallest normal subgroup such that the factor group is a $p$-group

Published on 28 Mar 2026 - 14:55

score 35 · Answer 10 · 2022-08-05 14:07:33

35

Views

Normality of $\frac{G_{i}\cap N}{G_{i+1}\cap N}$ in $\frac{G_i}{G_{i+1}}$ related to a composition series

Published on 05 Aug 2022 - 14:07

score 39 · Answer 11 · 2022-08-08 19:18:05

39

Views

Let $I = \langle x^2-3\rangle \subset \Bbb Q[x]$ and let $f=x-1 \in \Bbb Q[x]$. Show that $f+I$ is invertible in $\Bbb Q[x]/I$ and find the inverse.

Published on 08 Aug 2022 - 19:18

score 278 · Answer 12 · 2022-08-09 20:23:54

278

Views

If $G/M$ and $G/N$ both are cyclic and $\gcd(|M|,|N|)=1$, then $G$ is abelian.

Published on 09 Aug 2022 - 20:23

score 136 · Answer 13 · 2026-03-27 04:57:13

136

Views

Express $\mathbb{Z}^2/B$ as a direct product of cyclic groups

Published on 27 Mar 2026 - 4:57

score 151 · Answer 14 · 2022-08-12 09:21:44

151

Views

If $G_1\cong G_2$, $H_1\triangleleft G_1$, $H_2 \triangleleft G_2$ and $G_1/H_1\cong G_2/H_2$, then is $H_1\cong H_2$?

Published on 12 Aug 2022 - 9:21

score 74 · Answer 15 · 2022-08-12 17:00:20

74

Views

Determination of order of cosets in a factor group of a finite abelian group.

Published on 12 Aug 2022 - 17:00

List Question

Given a finite group $G$ and $N$ a normal subgroup such that $N\cong K$ where $K$ is Klein group and $G/N\cong \Bbb{Z}/7\Bbb{Z}$, prove $N\leq Z(G)$

Show that $\Bbb Z_{12}/\langle[3]_{12}\rangle \cong \Bbb Z_p$ for some $p \in \Bbb N$.

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.

Finite order elements of quotient group ${G}/{G^m}$

Extension of a homomorphism

Factor group of a factor groups

On a proof that the quaternion $8$-group $Q$ mod its center is isomorphic to the Klein four group

Prove that $A_5$ does not have a subgroup of order 15, 20, nor 30 (without using simple group and Sylow's Theorem)

Smallest normal subgroup such that the factor group is a $p$-group

Normality of $\frac{G_{i}\cap N}{G_{i+1}\cap N}$ in $\frac{G_i}{G_{i+1}}$ related to a composition series

Let $I = \langle x^2-3\rangle \subset \Bbb Q[x]$ and let $f=x-1 \in \Bbb Q[x]$. Show that $f+I$ is invertible in $\Bbb Q[x]/I$ and find the inverse.

If $G/M$ and $G/N$ both are cyclic and $\gcd(|M|,|N|)=1$, then $G$ is abelian.

Express $\mathbb{Z}^2/B$ as a direct product of cyclic groups

If $G_1\cong G_2$, $H_1\triangleleft G_1$, $H_2 \triangleleft G_2$ and $G_1/H_1\cong G_2/H_2$, then is $H_1\cong H_2$?

Determination of order of cosets in a factor group of a finite abelian group.

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