Question

$H$ 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 $G$.

score 789 · Answer 1 · 2026-04-13 17:43:28

789

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$H$ 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 13 Apr 2026 - 17:43

score 76 · Answer 2 · 2026-04-16 08:27:44

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

score 159 · Answer 3 · 2026-04-28 09:28:17

159

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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 28 Apr 2026 - 9:28

score 525 · Answer 4 · 2026-04-10 08:03:13

525

Views

What does the multiplication in cosets mean?

Published on 10 Apr 2026 - 8:03

score 640 · Answer 5 · 2026-04-10 21:59:24

640

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Given that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then $H$ is normal subgroup of $G$

Published on 10 Apr 2026 - 21:59

score 116 · Answer 6 · 2026-04-11 12:51:20

116

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Is G/H a group even if H is a subgroup of G but not a normal subgroup?

Published on 11 Apr 2026 - 12:51

score 2.4k · Answer 7 · 2026-05-10 21:47:32

2.4k

Views

Quotient ring local if Ring is local

Published on 10 May 2026 - 21:47

score 36 · Answer 8 · 2026-04-16 04:14:29

36

Views

Normal Subgroups and Quotient Groups help

Published on 16 Apr 2026 - 4:14

score 413 · Answer 9 · 2026-05-10 16:20:08

413

Views

Quotient group of the dihedral group by $\langle r^2 \rangle.$

Published on 10 May 2026 - 16:20

score 679 · Answer 10 · 2026-05-10 20:32:59

679

Views

Proving 2 quotient rings are isomorphic

Published on 10 May 2026 - 20:32

score 153 · Answer 11 · 2026-04-13 21:54:45

153

Views

Which group is this quotient group isomorphic to?

Published on 13 Apr 2026 - 21:54

score 107 · Answer 12 · 2026-04-14 17:03:00

107

Views

Find group isomorphic to the quotient group

Published on 14 Apr 2026 - 17:03

score 127 · Answer 13 · 2026-04-15 07:07:58

127

Views

Quotient cancellation for invertible ideals of orders in quadratic fields

Published on 15 Apr 2026 - 7:07

score 1.1k · Answer 14 · 2026-04-14 20:39:56

1.1k

Views

Quotient Group Is Finite Implies That Group Is Finite?

Published on 14 Apr 2026 - 20:39

score 72 · Answer 15 · 2026-04-12 07:51:46

72

Views

Elementary question about well-definedness of induced homomorphism

Published on 12 Apr 2026 - 7:51

List Question

$H$ 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 $G$.

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

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.

What does the multiplication in cosets mean?

Given that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then $H$ is normal subgroup of $G$

Is G/H a group even if H is a subgroup of G but not a normal subgroup?

Quotient ring local if Ring is local

Normal Subgroups and Quotient Groups help

Quotient group of the dihedral group by $\langle r^2 \rangle.$

Proving 2 quotient rings are isomorphic

Which group is this quotient group isomorphic to?

Find group isomorphic to the quotient group

Quotient cancellation for invertible ideals of orders in quadratic fields

Quotient Group Is Finite Implies That Group Is Finite?

Elementary question about well-definedness of induced homomorphism

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