# Questions tagged [combinatorial-group-theory]

Use this tag for questions about free groups and presentations of a group by generators and relations.

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### Using combinational group theoretical perspective on semidirect products, show $\langle r,s\mid r^8, s^2, srs=r^3\rangle$ has two Klein four subgroups

Note: This is an alternative-proof question, since I know how to prove the result but I'm asking for a particular kind of proof. Why? For the fun of it! Motivation: I've been trying to give a reason ...
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### Non-cyclic subgroup of order 4 in non-dihedral group

A group $G$ has sixteen elements: $$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$ where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$. (Note that $G$ is not a dihedral ...
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### Is there an enumeration of finitely presented groups?

I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a ...
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### Trying to find the set of unique representatives for the geodesics in the group $\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$

I am studying the conjugacy growth of the groups, and I encountered the following group: $$G=\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$$ Thanks to Derek for pointing out that $G$ is an ...
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### $\text{GL }_2(\mathbb{Z})$ as an HNN extension

This question arises from section $2$, exercise $13.8$ of Bogopolski's Introduction to Group Theory. I managed to show that $\text{GL}_2(\mathbb{Z})\cong D_4 *_{D_2} D_6$. Now, I want to take a random ...
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### Number of groups with a bounded short presentation

How many groups there are (up to isomorphism) with a presentation with at most $n$ generators and with relators of length at most $3$? I don't expect there exist a sharp solution, since I know that ...
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### Proving certain triangle groups are infinite

[Cross-posted to MathOverflow] Consider the Von Dyck group $$G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle$$ where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family ...
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### Prove that the group $G=\langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1\rangle$ is isomorphic to $S_4$

Prove that the group $$G = \langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1 \rangle$$ is isomorphic to $S_4$. Obviously, take $a=(12),\, b=(23),\, c=(34)$ and use Von Dyck's Theorem ...
### What is the group $G=\langle a,b \mid ab^2=b^3a,ba^3=a^2b\rangle$ [duplicate]
I feel that $G$ is infinite, however I don't know how to deal with the relationship $ab^2=b^3a,ba^3=a^2b$