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 · 2026-03-25 17:42:04

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 25 Mar 2026 - 17:42

score 242 · Answer 2 · 2018-10-04 04:24:29

242

Views

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

Published on 04 Oct 2018 - 4:24

score 2.2k · Answer 3 · 2026-03-26 01:12:13

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 26 Mar 2026 - 1:12

score 80 · Answer 4 · 2018-10-07 16:10:00

80

Views

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

Published on 07 Oct 2018 - 16:10

score 328 · Answer 5 · 2018-10-07 19:54:28

328

Views

Discrete normal subgroup of a connected linear lie group

Published on 07 Oct 2018 - 19:54

score 197 · Answer 6 · 2026-03-25 12:32:40

197

Views

Validation for a conjecture about Chinese Remainder Theorem for groups

Published on 25 Mar 2026 - 12:32

score 132 · Answer 7 · 2026-03-25 11:04:38

132

Views

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

Published on 25 Mar 2026 - 11:04

score 830 · Answer 8 · 2018-10-14 13:20:33

830

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 14 Oct 2018 - 13:20

score 467 · Answer 9 · 2026-03-25 22:04:09

467

Views

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

Published on 25 Mar 2026 - 22:04

score 494 · Answer 10 · 2026-03-25 12:50:49

494

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 25 Mar 2026 - 12:50

score 68 · Answer 11 · 2026-03-26 01:18:02

68

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 26 Mar 2026 - 1:18

score 33 · Answer 12 · 2018-10-17 08:16:12

33

Views

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

Published on 17 Oct 2018 - 8:16

score 44 · Answer 13 · 2026-03-25 23:38:28

44

Views

Question about Quotient of Product of Subgroups

Published on 25 Mar 2026 - 23:38

score 42 · Answer 14 · 2018-10-20 18:25:40

42

Views

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

Published on 20 Oct 2018 - 18:25

score 45 · Answer 15 · 2018-10-21 02:01:26

45

Views

Multiplication of Subgroups and Internal Direct Products

Published on 21 Oct 2018 - 2:01

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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