Question

isomorphism of normal subgroup

score 38 · Answer 1 · 2026-04-15 09:34:01

38

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isomorphism of normal subgroup

Published on 15 Apr 2026 - 9:34

score 55 · Answer 2 · 2026-04-09 11:56:28

55

Views

Is H a normal subgroup in G?

Published on 09 Apr 2026 - 11:56

score 122 · Answer 3 · 2026-04-17 07:40:40

122

Views

Prove $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$ where $G=MN$ and $M,N\triangleleft G$

Published on 17 Apr 2026 - 7:40

score 2.1k · Answer 4 · 2026-05-11 03:30:58

2.1k

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Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H).

Published on 11 May 2026 - 3:30

score 71 · Answer 5 · 2026-04-14 04:14:40

71

Views

Normal Subgroup Representation Theory

Published on 14 Apr 2026 - 4:14

score 463 · Answer 6 · 2026-05-11 08:01:53

463

Views

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.

Published on 11 May 2026 - 8:01

score 51 · Answer 7 · 2026-04-15 04:37:58

51

Views

Suppose that $H\trianglelefteq G$ and $H\leq K\leq G$. Show $H\trianglelefteq K$ and that $K/H\leq G/H$

Published on 15 Apr 2026 - 4:37

score 402 · Answer 8 · 2026-04-10 06:13:25

402

Views

How do generators of a group work?

Published on 10 Apr 2026 - 6:13

score 790 · Answer 9 · 2026-05-11 05:07:46

790

Views

$H$ is a subgroup of a finite group $G$ such that $|H|$ and $\big([G:H]-1\big)!$ are relatively prime. Prove that $H$ is normal in $G$.

Published on 11 May 2026 - 5:07

score 72 · Answer 10 · 2026-04-11 06:15:09

72

Views

Does $G \times H = K$ iff $G$ and $H$ are normal in $K$?

Published on 11 Apr 2026 - 6:15

score 188 · Answer 11 · 2026-05-10 12:39:28

188

Views

Normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially; What is wrong with the following trivial argument?

Published on 10 May 2026 - 12:39

score 288 · Answer 12 · 2026-05-10 14:59:48

288

Views

Are all verbal automorphisms inner power automorphisms?

Published on 10 May 2026 - 14:59

score 66 · Answer 13 · 2026-04-15 13:01:15

66

Views

Normal groups and homomorphism are the same, and this gives an approach to isomorphism theorem?

Published on 15 Apr 2026 - 13:01

score 1.9k · Answer 14 · 2026-04-15 07:04:09

1.9k

Views

Equivalent definitions of normal subgroups

Published on 15 Apr 2026 - 7:04

score 159 · Answer 15 · 2026-04-28 09:28:17

159

Views

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent.

Published on 28 Apr 2026 - 9:28

List Question

isomorphism of normal subgroup

Is H a normal subgroup in G?

Prove $G/(M\cap N) \cong M/(M\cap N) \times N/(M\cap N)$ where $G=MN$ and $M,N\triangleleft G$

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H).

Normal Subgroup Representation Theory

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$.

Suppose that $H\trianglelefteq G$ and $H\leq K\leq G$. Show $H\trianglelefteq K$ and that $K/H\leq G/H$

How do generators of a group work?

$H$ is a subgroup of a finite group $G$ such that $|H|$ and $\big([G:H]-1\big)!$ are relatively prime. Prove that $H$ is normal in $G$.

Does $G \times H = K$ iff $G$ and $H$ are normal in $K$?

Normal subgroup $N$ of a p-group $G$ intersects $Z(G)$ nontrivially; What is wrong with the following trivial argument?

Are all verbal automorphisms inner power automorphisms?

Normal groups and homomorphism are the same, and this gives an approach to isomorphism theorem?

Equivalent definitions of normal subgroups

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent.

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