Question

Product of subgroup and normal subgroup with prime index

score 382 · Answer 1 · 2020-06-21 16:30:18

382

Views

Product of subgroup and normal subgroup with prime index

Published on 21 Jun 2020 - 16:30

score 2.9k · Answer 2 · 2020-06-21 22:21:58

2.9k

Views

Let $G$ be a finite group, $H\le G$ and $N\unlhd G$ such that $\gcd(|H|,[G:N])=1$. Prove that $H\le N$.

Published on 21 Jun 2020 - 22:21

score 1.4k · Answer 3 · 2026-03-29 18:33:31

1.4k

Views

Prove that $K = \{ (1); (1 2)(3 4); (1 3)(2 4); (1 4)(2 3) \} \subseteq S_4$ is a normal subgroup of $A_4$ and $S_4$

Published on 29 Mar 2026 - 18:33

score 1.5k · Answer 4 · 2026-03-26 17:32:18

1.5k

Views

Let $H\triangleleft G$. Prove that $G/H$ is abelian iff $ [G, G] \subseteq H$

Published on 26 Mar 2026 - 17:32

score 686 · Answer 5 · 2020-06-25 06:18:24

686

Views

Suppose $H<G$, let $N=\bigcap_{x\in G} xHx^{-1}$, show $N$ is the largest normal subgroup of $G$ contained in $H$.

Published on 25 Jun 2020 - 6:18

score 460 · Answer 6 · 2026-03-26 17:32:28

460

Views

The quotient of a quotient group by another quotient group

Published on 26 Mar 2026 - 17:32

score 266 · Answer 7 · 2026-03-28 07:57:11

266

Views

Intersection of Sylow 3-subgroups in a group of order 189.

Published on 28 Mar 2026 - 7:57

score 318 · Answer 8 · 2020-06-29 00:33:24

318

Views

Showing $f(h)=g^{-1}h^{-1}gh$ for any $g \in G, h\in H$ where $H$ is normal subgroup of $G$.

Published on 29 Jun 2020 - 0:33

score 1.1k · Answer 9 · 2020-07-01 18:54:00

1.1k

Views

Group of order 28 with normal subgroup of order 4 is abelian

Published on 01 Jul 2020 - 18:54

score 40 · Answer 10 · 2020-07-03 08:30:00

40

Views

If $A, B, H \leq G$ such that $A \triangleleft B$ and $H \triangleleft G$, then $HA \triangleleft HB$

Published on 03 Jul 2020 - 8:30

score 138 · Answer 11 · 2026-03-25 14:23:38

138

Views

Homogeneous space and Uniform Subgroup of Lie groups

Published on 25 Mar 2026 - 14:23

score 185 · Answer 12 · 2026-03-29 18:33:34

185

Views

Let $G$ be a group of order $2016 = 2^5 \cdot 3^2 \cdot 7$ in which all elements of order $7$ are conjugate.

Published on 29 Mar 2026 - 18:33

score 1k · Answer 13 · 2020-07-05 09:51:35

1k

Views

Is $G^2$ necessarily a subgroup of $G$?

Published on 05 Jul 2020 - 9:51

score 34 · Answer 14 · 2020-07-05 20:19:32

34

Views

Let $K'<K<G$ be a chain of groups s.t. $K\subset N_G(K')$. Is that true that $N_G(K)=N_G(K')$?

Published on 05 Jul 2020 - 20:19

score 30 · Answer 15 · 2020-07-06 11:17:17

30

Views

Proving the correspondence theorem for groups without the inverse map

Published on 06 Jul 2020 - 11:17

List Question

Product of subgroup and normal subgroup with prime index

Let $G$ be a finite group, $H\le G$ and $N\unlhd G$ such that $\gcd(|H|,[G:N])=1$. Prove that $H\le N$.

Prove that $K = \{ (1); (1 2)(3 4); (1 3)(2 4); (1 4)(2 3) \} \subseteq S_4$ is a normal subgroup of $A_4$ and $S_4$

Let $H\triangleleft G$. Prove that $G/H$ is abelian iff $ [G, G] \subseteq H$

Suppose $H<G$, let $N=\bigcap_{x\in G} xHx^{-1}$, show $N$ is the largest normal subgroup of $G$ contained in $H$.

The quotient of a quotient group by another quotient group

Intersection of Sylow 3-subgroups in a group of order 189.

Showing $f(h)=g^{-1}h^{-1}gh$ for any $g \in G, h\in H$ where $H$ is normal subgroup of $G$.

Group of order 28 with normal subgroup of order 4 is abelian

If $A, B, H \leq G$ such that $A \triangleleft B$ and $H \triangleleft G$, then $HA \triangleleft HB$

Homogeneous space and Uniform Subgroup of Lie groups

Let $G$ be a group of order $2016 = 2^5 \cdot 3^2 \cdot 7$ in which all elements of order $7$ are conjugate.

Is $G^2$ necessarily a subgroup of $G$?

Let $K'<K<G$ be a chain of groups s.t. $K\subset N_G(K')$. Is that true that $N_G(K)=N_G(K')$?

Proving the correspondence theorem for groups without the inverse map

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