Question

Let $H\le G$ s.t. $x^2\in H$ for all $x\in G$. Prove that $H\unlhd G$ and $G/H$ is abelian.

score 1.5k · Answer 1 · 2026-03-26 04:34:58

1.5k

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Let $H\le G$ s.t. $x^2\in H$ for all $x\in G$. Prove that $H\unlhd G$ and $G/H$ is abelian.

Published on 26 Mar 2026 - 4:34

score 1.8k · Answer 2 · 2019-11-21 23:20:32

1.8k

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Showing that Klein Four Group is a normal subgroup of $S_4$

Published on 21 Nov 2019 - 23:20

score 219 · Answer 3 · 2026-03-26 01:03:30

219

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Isomorphism of two simple normal subgroups of a group $G$, given that $G$ has composition series

Published on 26 Mar 2026 - 1:03

score 88 · Answer 4 · 2019-11-22 09:24:18

88

Views

Commensurability and normal subgroups

Published on 22 Nov 2019 - 9:24

score 247 · Answer 5 · 2026-03-26 04:31:15

247

Views

What is the relation between a group action and its restriction to a normal subgroup?

Published on 26 Mar 2026 - 4:31

score 473 · Answer 6 · 2026-03-25 17:19:21

473

Views

A Normal subgroup of a free group generated via a fixed integer

Published on 25 Mar 2026 - 17:19

score 350 · Answer 7 · 2026-03-26 19:20:53

350

Views

Show that the Sylow 11-subgroup of a group of order 990 is normal

Published on 26 Mar 2026 - 19:20

score 136 · Answer 8 · 2019-11-25 17:55:20

136

Views

Largest normal subgroup with self-centralizing Fitting subgroups

Published on 25 Nov 2019 - 17:55

score 387 · Answer 9 · 2019-11-25 21:46:58

387

Views

A group of order 21 has conjugacy classes of sizes 1, 3, 3, 7, and 7, and it has a normal subgroup of order 7. Find its character table.

Published on 25 Nov 2019 - 21:46

score 67 · Answer 10 · 2026-03-27 16:47:50

67

Views

If $\left| G \right| = p^aq^b$, then $G$ has a normal Sylow $p$-subgroup.

Published on 27 Mar 2026 - 16:47

score 143 · Answer 11 · 2026-03-25 17:39:22

143

Views

Show that if $H$ is a normal subgroup of $G$

Published on 25 Mar 2026 - 17:39

score 169 · Answer 12 · 2026-03-22 07:24:35

169

Views

Cayley graph of a quotient

Published on 22 Mar 2026 - 7:24

score 53 · Answer 13 · 2026-03-27 04:38:09

53

Views

If $\langle a \rangle \unlhd \langle a,b \rangle$ and $a,b$ have finite order, then $\langle a, b \rangle$ is finite.

Published on 27 Mar 2026 - 4:38

score 353 · Answer 14 · 2019-11-30 08:52:01

353

Views

Prove that, when $N$ is a normal subgroup, $(gN)^a=(g^a)N$ for all $a \in \mathbb{Z}$.

Published on 30 Nov 2019 - 8:52

score 304 · Answer 15 · 2019-11-30 20:16:07

304

Views

Prove that $k h = h k$ for $h \in H $ & $k \in K $

Published on 30 Nov 2019 - 20:16

List Question

Let $H\le G$ s.t. $x^2\in H$ for all $x\in G$. Prove that $H\unlhd G$ and $G/H$ is abelian.

Showing that Klein Four Group is a normal subgroup of $S_4$

Isomorphism of two simple normal subgroups of a group $G$, given that $G$ has composition series

Commensurability and normal subgroups

What is the relation between a group action and its restriction to a normal subgroup?

A Normal subgroup of a free group generated via a fixed integer

Show that the Sylow 11-subgroup of a group of order 990 is normal

Largest normal subgroup with self-centralizing Fitting subgroups

A group of order 21 has conjugacy classes of sizes 1, 3, 3, 7, and 7, and it has a normal subgroup of order 7. Find its character table.

If $\left| G \right| = p^aq^b$, then $G$ has a normal Sylow $p$-subgroup.

Show that if $H$ is a normal subgroup of $G$

Cayley graph of a quotient

If $\langle a \rangle \unlhd \langle a,b \rangle$ and $a,b$ have finite order, then $\langle a, b \rangle$ is finite.

Prove that, when $N$ is a normal subgroup, $(gN)^a=(g^a)N$ for all $a \in \mathbb{Z}$.

Prove that $k h = h k$ for $h \in H $ & $k \in K $

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