List questions from combinatorial-group-theory

Published on 10 May 2026 - 20:48

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$G$ abelian $\{ a^ib^j : a^m = b^n = 1, a^2=b^3 \}$, prove that $G=\langle c:c^k=1 \rangle$

Published on 25 Mar 2026 - 20:20

169

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Find a presentation with generators $g = (1,2)$ and $h = (2, 3)$ and three relations.

Published on 25 Mar 2026 - 20:18

81

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Do there exist characteristic subgroups, that are not quasiverbal?

Published on 09 Apr 2026 - 9:30

177

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Kernel of group epimorphism with finitely presented is the normal closure of a finite subset .

Published on 11 May 2026 - 3:30

71

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Show that $\langle s,t \mid t^2, tst^{-1}s \rangle \cong \langle a,b \mid a^2, b^2 \rangle$.

Published on 25 Mar 2026 - 20:19

58

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The relation $x\sim y$ iff $S(x)=S(y)$ for $S(x)=\{z\in G:\langle x,z\rangle=G\}$ for a finite, non-cyclic group $G$.

Published on 25 Mar 2026 - 20:22

178

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An infinite group generated by all its order two elements

Published on 10 May 2026 - 16:04

260

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$PSL(2, \Bbb{Z})\cong \Bbb{Z}_2 \ast \Bbb{Z}_3$

Published on 25 Mar 2026 - 20:18

1.5k

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On proving that a finitely generated group has a finite number of subgroups with index $n$.

Published on 26 Apr 2026 - 7:50

573

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finite index subgroups in free group non-trivial intersection with each of the non-trivial subgroups of the free group.

Published on 24 Apr 2026 - 22:30

95

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Determining whether an element of a free product of cyclic groups is a commutator.

Published on 10 May 2026 - 16:57

275

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Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, are there classes of such groups where this is known?

Published on 25 Mar 2026 - 20:22

102

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Free groups. Help me to understand this theorem.

Published on 25 Mar 2026 - 20:17

361

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How do you find a $(21,5,1)$-difference set in $(\mathbb{Z}_{21}, +)$?

Published on 10 May 2026 - 18:23

List Question

On free abelian groups

$G$ abelian $\{ a^ib^j : a^m = b^n = 1, a^2=b^3 \}$, prove that $G=\langle c:c^k=1 \rangle$

Find a presentation with generators $g = (1,2)$ and $h = (2, 3)$ and three relations.

Do there exist characteristic subgroups, that are not quasiverbal?

Kernel of group epimorphism with finitely presented is the normal closure of a finite subset .

Show that $\langle s,t \mid t^2, tst^{-1}s \rangle \cong \langle a,b \mid a^2, b^2 \rangle$.

The relation $x\sim y$ iff $S(x)=S(y)$ for $S(x)=\{z\in G:\langle x,z\rangle=G\}$ for a finite, non-cyclic group $G$.

An infinite group generated by all its order two elements

$PSL(2, \Bbb{Z})\cong \Bbb{Z}_2 \ast \Bbb{Z}_3$

On proving that a finitely generated group has a finite number of subgroups with index $n$.

finite index subgroups in free group non-trivial intersection with each of the non-trivial subgroups of the free group.

Determining whether an element of a free product of cyclic groups is a commutator.

Is there a criterion for which $BS(m,n)$ are solvable (and non-solvable)? If not, are there classes of such groups where this is known?

Free groups. Help me to understand this theorem.

How do you find a $(21,5,1)$-difference set in $(\mathbb{Z}_{21}, +)$?

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