Question

Does there exist a group that is both a free product and a direct product of nontrivial groups?

score 520 · Answer 1 · 2019-02-12 20:25:53

520

Views

Does there exist a group that is both a free product and a direct product of nontrivial groups?

Published on 12 Feb 2019 - 20:25

score 114 · Answer 2 · 2026-03-22 09:10:42

114

Views

Quaternion Group: Determine that $i^4 = 1$.

Published on 22 Mar 2026 - 9:10

score 230 · Answer 3 · 2026-03-22 09:32:40

230

Views

Can I conclude that my group is finitely generated, if it is a homomorphic image of a free-group on finitely many generators?

Published on 22 Mar 2026 - 9:32

score 134 · Answer 4 · 2026-03-25 07:48:41

134

Views

Proving the Free Abelian Group is Free Abelian...?

Published on 25 Mar 2026 - 7:48

score 77 · Answer 5 · 2026-03-22 04:39:23

77

Views

Reference for a result in group theory

Published on 22 Mar 2026 - 4:39

score 137 · Answer 6 · 2026-03-22 01:21:40

137

Views

If the inner automorphism group of a group $G$ is residually finite, then $G$ is residually finite group.

Published on 22 Mar 2026 - 1:21

score 295 · Answer 7 · 2026-03-21 20:20:57

295

Views

If we are handed the presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group?

Published on 21 Mar 2026 - 20:20

score 198 · Answer 8 · 2026-03-21 00:54:45

198

Views

Fixed points of a group action on tree

Published on 21 Mar 2026 - 0:54

score 111 · Answer 9 · 2026-03-22 11:25:01

111

Views

Homomorphism condition for a subset of a group generates the whole group

Published on 22 Mar 2026 - 11:25

score 60 · Answer 10 · 2019-03-08 16:38:25

60

Views

A finite group $G$ and fixed $k\geq 1$ where for every $n\geq 1$, the $n$-direct product $G^n=G\times\dots\times G$ is $k$-generated?

Published on 08 Mar 2019 - 16:38

score 717 · Answer 11 · 2026-03-22 10:06:08

717

Views

How to show that $|D_{2n}| = 2n$ via the presentation?

Published on 22 Mar 2026 - 10:06

score 35 · Answer 12 · 2026-03-21 17:35:58

35

Views

Symmetrising the relations in a presentation of a group

Published on 21 Mar 2026 - 17:35

score 349 · Answer 13 · 2026-03-21 20:35:33

349

Views

Reduced word is uniquely written

Published on 21 Mar 2026 - 20:35

score 189 · Answer 14 · 2026-03-21 19:22:24

189

Views

$\infty$-ended subgroups of one-ended groups

Published on 21 Mar 2026 - 19:22

score 241 · Answer 15 · 2026-03-22 03:40:03

241

Views

Proving $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$

Published on 22 Mar 2026 - 3:40

List Question

Does there exist a group that is both a free product and a direct product of nontrivial groups?

Quaternion Group: Determine that $i^4 = 1$.

Can I conclude that my group is finitely generated, if it is a homomorphic image of a free-group on finitely many generators?

Proving the Free Abelian Group is Free Abelian...?

Reference for a result in group theory

If the inner automorphism group of a group $G$ is residually finite, then $G$ is residually finite group.

If we are handed the presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group?

Fixed points of a group action on tree

Homomorphism condition for a subset of a group generates the whole group

A finite group $G$ and fixed $k\geq 1$ where for every $n\geq 1$, the $n$-direct product $G^n=G\times\dots\times G$ is $k$-generated?

How to show that $|D_{2n}| = 2n$ via the presentation?

Symmetrising the relations in a presentation of a group

Reduced word is uniquely written

$\infty$-ended subgroups of one-ended groups

Proving $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$

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