Question

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

score 469 · Answer 1 · 2026-03-26 09:18:08

469

Views

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

Published on 26 Mar 2026 - 9:18

score 119 · Answer 2 · 2019-12-30 01:27:31

119

Views

Counterexamples for $gh=h'g$ for some $h'\in H$ doesn't implies $gh=hg$.

Published on 30 Dec 2019 - 1:27

score 231 · Answer 3 · 2019-12-31 05:32:11

231

Views

Group with exactly $2$ elements of order $6$ has a normal subgroup of order $3$

Published on 31 Dec 2019 - 5:32

score 183 · Answer 4 · 2020-01-01 21:32:11

183

Views

Given $G$ is a finite group, $a$ and $b \in G$, $H\le G$, and $N=N(H)$, prove that if $aHa^{-1}=bHb^{-1}$, then $aN=bN$.

Published on 01 Jan 2020 - 21:32

score 207 · Answer 5 · 2020-01-06 20:53:44

207

Views

Refinement of a Jordan-Hölder series of a group $G$

Published on 06 Jan 2020 - 20:53

score 147 · Answer 6 · 2026-03-26 03:12:46

147

Views

Presentation $\langle x,y\mid x^3=y^3=(xy)^3=1\rangle $ with normal abelian subgroup.

Published on 26 Mar 2026 - 3:12

score 1k · Answer 7 · 2026-03-26 11:00:48

1k

Views

N contains the commutator subgroup if and only if G/N is abelian, proof by words?

Published on 26 Mar 2026 - 11:00

score 499 · Answer 8 · 2026-03-26 09:18:06

499

Views

Verifying my proposed proof: If $(G:H)=m$, the order of every element of $G/H$ is a divisor of $m$

Published on 26 Mar 2026 - 9:18

score 59 · Answer 9 · 2020-01-08 08:22:21

59

Views

Coset representatives of a subgroup

Published on 08 Jan 2020 - 8:22

score 49 · Answer 10 · 2020-01-08 18:21:53

49

Views

Quotients of a composition series of a group G

Published on 08 Jan 2020 - 18:21

score 1.2k · Answer 11 · 2026-03-26 01:08:03

1.2k

Views

Rigorous proof of the Jordan-Hölder Theorem

Published on 26 Mar 2026 - 1:08

score 327 · Answer 12 · 2020-01-09 23:54:58

327

Views

Why do normal subgroups "work"?

Published on 09 Jan 2020 - 23:54

score 511 · Answer 13 · 2020-01-11 11:58:53

511

Views

Is a group a normal subgroup of itself?

Published on 11 Jan 2020 - 11:58

score 290 · Answer 14 · 2026-03-25 21:47:23

290

Views

Group action by a semi-direct product.

Published on 25 Mar 2026 - 21:47

score 717 · Answer 15 · 2020-01-13 08:16:31

717

Views

Normal subgroups of infinite groups

Published on 13 Jan 2020 - 8:16

List Question

How to show that the dihedral group $D_{2\cdot 8}$ is the quotient of the free group on $2$ generators by a certain normal subgroup?

Counterexamples for $gh=h'g$ for some $h'\in H$ doesn't implies $gh=hg$.

Group with exactly $2$ elements of order $6$ has a normal subgroup of order $3$

Given $G$ is a finite group, $a$ and $b \in G$, $H\le G$, and $N=N(H)$, prove that if $aHa^{-1}=bHb^{-1}$, then $aN=bN$.

Refinement of a Jordan-Hölder series of a group $G$

Presentation $\langle x,y\mid x^3=y^3=(xy)^3=1\rangle $ with normal abelian subgroup.

N contains the commutator subgroup if and only if G/N is abelian, proof by words?

Verifying my proposed proof: If $(G:H)=m$, the order of every element of $G/H$ is a divisor of $m$

Coset representatives of a subgroup

Quotients of a composition series of a group G

Rigorous proof of the Jordan-Hölder Theorem

Why do normal subgroups "work"?

Is a group a normal subgroup of itself?

Group action by a semi-direct product.

Normal subgroups of infinite groups

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