Question

isomorphism of normal subgroup

score 35 · Answer 1 · 2018-12-02 21:38:02

35

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isomorphism of normal subgroup

Published on 02 Dec 2018 - 21:38

score 52 · Answer 2 · 2018-12-03 16:17:22

52

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Is H a normal subgroup in G?

Published on 03 Dec 2018 - 16:17

score 119 · Answer 3 · 2018-12-03 19:00:43

119

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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 03 Dec 2018 - 19:00

score 2.1k · Answer 4 · 2026-03-25 12:50:56

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 25 Mar 2026 - 12:50

score 68 · Answer 5 · 2018-12-08 20:46:44

68

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Normal Subgroup Representation Theory

Published on 08 Dec 2018 - 20:46

score 460 · Answer 6 · 2026-03-25 14:25:40

460

Views

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

Published on 25 Mar 2026 - 14:25

score 48 · Answer 7 · 2018-12-09 22:33:35

48

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

Published on 09 Dec 2018 - 22:33

score 399 · Answer 8 · 2018-12-10 04:16:50

399

Views

How do generators of a group work?

Published on 10 Dec 2018 - 4:16

score 787 · Answer 9 · 2026-03-25 14:35:51

787

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 25 Mar 2026 - 14:35

score 69 · Answer 10 · 2018-12-13 23:46:12

69

Views

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

Published on 13 Dec 2018 - 23:46

score 185 · Answer 11 · 2026-03-25 06:13:03

185

Views

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

Published on 25 Mar 2026 - 6:13

score 283 · Answer 12 · 2026-03-25 06:00:37

283

Views

Are all verbal automorphisms inner power automorphisms?

Published on 25 Mar 2026 - 6:00

score 65 · Answer 13 · 2026-03-26 13:01:19

65

Views

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

Published on 26 Mar 2026 - 13:01

score 1.9k · Answer 14 · 2018-12-17 13:26:49

1.9k

Views

Equivalent definitions of normal subgroups

Published on 17 Dec 2018 - 13:26

score 154 · Answer 15 · 2026-03-22 11:51:36

154

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 22 Mar 2026 - 11:51

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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