Question

Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

score 83 · Answer 1 · 2022-04-21 17:00:01

83

Views

Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

Published on 21 Apr 2022 - 17:00

score 223 · Answer 2 · 2022-04-22 13:44:52

223

Views

Cyclic quotient group as a subgroup?

Published on 22 Apr 2022 - 13:44

score 95 · Answer 3 · 2026-03-28 06:40:33

95

Views

How to show that if $G/N$ and $N$ soluble then so is $G$

Published on 28 Mar 2026 - 6:40

score 36 · Answer 4 · 2022-04-24 19:12:09

36

Views

Existence of well defined map $R/J\to R/I$ implies $J\cong M\subseteq I$?

Published on 24 Apr 2022 - 19:12

score 44 · Answer 5 · 2022-04-27 15:58:06

44

Views

What does this proposition state regarding normal subgroups?

Published on 27 Apr 2022 - 15:58

score 49 · Answer 6 · 2022-05-01 08:10:29

Number of cosets and the their representations in quotient groups of the form $(Z\times Z)/\langle (a,b) \rangle$ for different pairs of $(a,b)$

score 81 · Answer 7 · 2022-05-02 20:09:11

81

Views

Let $G$ be a group with $K\unlhd G$ and Abelian $N\unlhd G$. Is $G/(NK)$ an Abelian group?

Published on 02 May 2022 - 20:09

score 215 · Answer 8 · 2022-05-05 22:01:07

215

Views

Example of $H$, $N$ two normal subgroups of a group G such that $G/N \cong G/H$ but $N \ncong H$

Published on 05 May 2022 - 22:01

score 131 · Answer 9 · 2022-05-07 07:01:10

131

Views

Proving that $\mathbb{R}/\mathbb{Z} \cong S^1$ as groups

Published on 07 May 2022 - 7:01

score 599 · Answer 10 · 2026-03-29 03:36:25

599

Views

$H \unlhd G$, $\pi:G \to H$ is a group homomorphism with $\pi(h)=h$. Show that $G \cong H × G/H$

Published on 29 Mar 2026 - 3:36

score 100 · Answer 11 · 2026-03-27 21:24:21

100

Views

Index of Socle of primitive permutation group $G$

Published on 27 Mar 2026 - 21:24

score 69 · Answer 12 · 2022-05-12 10:49:29

69

Views

Question about the solution to the exercise: prove that $\mathbb{Q}/\mathbb{Z}$ contains elements of every possible finite order

Published on 12 May 2022 - 10:49

score 94 · Answer 13 · 2022-05-12 13:09:40

94

Views

Notation: What is $C(G)$?

Published on 12 May 2022 - 13:09

score 161 · Answer 14 · 2022-05-18 19:37:28

161

Views

Quotient of $GL(\mathbb{R}^n)$ by $O(\mathbb{R}^n)$

Published on 18 May 2022 - 19:37

score 79 · Answer 15 · 2026-03-28 21:51:43

79

Views

Solution check request for the question: If $G=AB$ and $A\cap B=N,$ prove that $G/N\cong A/N\times B/N\\$

Published on 28 Mar 2026 - 21:51

List Question

Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

Cyclic quotient group as a subgroup?

How to show that if $G/N$ and $N$ soluble then so is $G$

Existence of well defined map $R/J\to R/I$ implies $J\cong M\subseteq I$?

What does this proposition state regarding normal subgroups?

Number of cosets and the their representations in quotient groups of the form $(Z\times Z)/\langle (a,b) \rangle$ for different pairs of $(a,b)$

Let $G$ be a group with $K\unlhd G$ and Abelian $N\unlhd G$. Is $G/(NK)$ an Abelian group?

Example of $H$, $N$ two normal subgroups of a group G such that $G/N \cong G/H$ but $N \ncong H$

Proving that $\mathbb{R}/\mathbb{Z} \cong S^1$ as groups

$H \unlhd G$, $\pi:G \to H$ is a group homomorphism with $\pi(h)=h$. Show that $G \cong H × G/H$

Index of Socle of primitive permutation group $G$

Question about the solution to the exercise: prove that $\mathbb{Q}/\mathbb{Z}$ contains elements of every possible finite order

Notation: What is $C(G)$?

Quotient of $GL(\mathbb{R}^n)$ by $O(\mathbb{R}^n)$

Solution check request for the question: If $G=AB$ and $A\cap B=N,$ prove that $G/N\cong A/N\times B/N\\$

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