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 786 · Answer 1 · 2018-12-11 15:17:18

786

Views

$H$ 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 11 Dec 2018 - 15:17

score 73 · Answer 2 · 2018-12-13 08:23:31

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 154 · Answer 3 · 2026-03-22 11:51:36

154

Views

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 22 Mar 2026 - 11:51

score 522 · Answer 4 · 2018-12-28 22:23:16

522

Views

What does the multiplication in cosets mean?

Published on 28 Dec 2018 - 22:23

score 637 · Answer 5 · 2018-12-29 14:49:39

637

Views

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

Published on 29 Dec 2018 - 14:49

score 113 · Answer 6 · 2019-01-01 18:11:59

113

Views

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

Published on 01 Jan 2019 - 18:11

score 2.4k · Answer 7 · 2026-03-25 02:59:47

2.4k

Views

Quotient ring local if Ring is local

Published on 25 Mar 2026 - 2:59

score 33 · Answer 8 · 2019-01-06 16:34:35

33

Views

Normal Subgroups and Quotient Groups help

Published on 06 Jan 2019 - 16:34

score 409 · Answer 9 · 2026-03-25 06:05:52

409

Views

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

Published on 25 Mar 2026 - 6:05

score 675 · Answer 10 · 2019-01-18 20:49:52

675

Views

Proving 2 quotient rings are isomorphic

Published on 18 Jan 2019 - 20:49

score 150 · Answer 11 · 2019-01-19 18:26:17

150

Views

Which group is this quotient group isomorphic to?

Published on 19 Jan 2019 - 18:26

score 104 · Answer 12 · 2019-01-26 23:01:54

104

Views

Find group isomorphic to the quotient group

Published on 26 Jan 2019 - 23:01

score 123 · Answer 13 · 2019-01-30 08:24:48

123

Views

Quotient cancellation for invertible ideals of orders in quadratic fields

Published on 30 Jan 2019 - 8:24

score 1.1k · Answer 14 · 2019-02-01 01:51:48

1.1k

Views

Quotient Group Is Finite Implies That Group Is Finite?

Published on 01 Feb 2019 - 1:51

score 69 · Answer 15 · 2019-02-04 15:46:59

69

Views

Elementary question about well-definedness of induced homomorphism

Published on 04 Feb 2019 - 15:46

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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