Question

Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$ then $G/H \cong K?$

score 1.1k · Answer 1 · 2026-05-10 15:52:36

1.1k

Views

Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$ then $G/H \cong K?$

Published on 10 May 2026 - 15:52

score 245 · Answer 2 · 2026-04-14 10:54:00

245

Views

Prove that if a subgroup $H$ of $G$ contains all of its conjugates, it is a normal subgroup.

Published on 14 Apr 2026 - 10:54

score 2.2k · Answer 3 · 2026-04-14 20:57:01

2.2k

Views

>$f : G → G′$ is a group homomorphism. (i). $H′$ is a normal subgroup of $G′$. Prove $f^-1(H′) = [ h ∈ G|f(h) ∈ H′] $ is a normal subgroup of $G$.

Published on 14 Apr 2026 - 20:57

score 83 · Answer 4 · 2026-04-16 01:29:11

83

Views

Prove the number of conjugates of K in L is $[Gal(L|F) : N]$ where $N$ is the normalizer subgroup

Published on 16 Apr 2026 - 1:29

score 331 · Answer 5 · 2026-04-12 22:05:48

331

Views

Discrete normal subgroup of a connected linear lie group

Published on 12 Apr 2026 - 22:05

score 198 · Answer 6 · 2026-05-10 20:33:14

198

Views

Validation for a conjecture about Chinese Remainder Theorem for groups

Published on 10 May 2026 - 20:33

score 135 · Answer 7 · 2026-05-10 22:10:08

135

Views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

Published on 10 May 2026 - 22:10

score 833 · Answer 8 · 2026-04-16 16:01:02

833

Views

In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

Published on 16 Apr 2026 - 16:01

score 469 · Answer 9 · 2026-04-10 21:12:41

469

Views

Partition of a conjugacy class to conjugacy classes of a normal subgroup

Published on 10 Apr 2026 - 21:12

score 498 · Answer 10 · 2026-05-10 21:11:10

498

Views

Let $P$ be a partition of a group with $AB \subseteq C$. Why is $1 \in P_n$? $P_n$ is the equivalence class of $n \in N$ and $1 \in N=P_1$.

Published on 10 May 2026 - 21:11

score 72 · Answer 11 · 2026-04-17 11:58:19

72

Views

Prove that $\varphi\big(C_G(x)\big) =C_H\big(\varphi(x)\big)$, where $\varphi:G\to H$ is a group homomorphism with certain properties.

Published on 17 Apr 2026 - 11:58

score 36 · Answer 12 · 2026-04-16 02:55:22

36

Views

Proving the bound $[G:Core(H)]\leq [G:H]!$

Published on 16 Apr 2026 - 2:55

score 46 · Answer 13 · 2026-04-11 17:44:55

46

Views

Question about Quotient of Product of Subgroups

Published on 11 Apr 2026 - 17:44

score 45 · Answer 14 · 2026-04-10 20:10:04

45

Views

Why require a normal subgroup to be also a subgroup, in the definition?

Published on 10 Apr 2026 - 20:10

score 48 · Answer 15 · 2026-04-13 08:43:13

48

Views

Multiplication of Subgroups and Internal Direct Products

Published on 13 Apr 2026 - 8:43

List Question

Is it true that if $H,K$ are two normal subgroups of $G$ such that $G=HK$ an $H\cap K = \{1\},$ then $G/H \cong K?$

Prove that if a subgroup $H$ of $G$ contains all of its conjugates, it is a normal subgroup.

>$f : G → G′$ is a group homomorphism. (i). $H′$ is a normal subgroup of $G′$. Prove $f^-1(H′) = [ h ∈ G|f(h) ∈ H′] $ is a normal subgroup of $G$.

Prove the number of conjugates of K in L is $[Gal(L|F) : N]$ where $N$ is the normalizer subgroup

Discrete normal subgroup of a connected linear lie group

Validation for a conjecture about Chinese Remainder Theorem for groups

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In proving G contains an element of order 15 if contains normal subgroups of orders 3 and 5, respectively, is $HK$ itself cyclic with order 15?

Partition of a conjugacy class to conjugacy classes of a normal subgroup

Let $P$ be a partition of a group with $AB \subseteq C$. Why is $1 \in P_n$? $P_n$ is the equivalence class of $n \in N$ and $1 \in N=P_1$.

Prove that $\varphi\big(C_G(x)\big) =C_H\big(\varphi(x)\big)$, where $\varphi:G\to H$ is a group homomorphism with certain properties.

Proving the bound $[G:Core(H)]\leq [G:H]!$

Question about Quotient of Product of Subgroups

Why require a normal subgroup to be also a subgroup, in the definition?

Multiplication of Subgroups and Internal Direct Products

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