# 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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### Expressing the fundamental group of the Klein bottle as an HNN extension appears to contradict Britton's Lemma

Express the Klein bottle group $G'=\langle T,A\mid ATAT^{-1}\rangle$ as an HNN extension of $\mathbb{Z}$ as follows (using notation from Wikipedia for convenience: https://en.wikipedia.org/wiki/...
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• 431
1 vote
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### Group actions on Cartesian Product of a path and a cycle

Let $G = C_m \square P_n$ (grid with $m$ rows, $n$ columns, where bottom and top row are connected via edges). What are all of the possible symmetries of $G$? Equivalently, I would like to describe ...
1 vote
29 views

### Reference request: proof of the Rips Construction

I'm trying to understand how the Rips Construction works. In particular, I'd like to understand why the presentation cooked up by the Rips construction (which if I'm not mistaken is not explicitly ...
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1 vote
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### Clarification on the Nielsen-Schreier Theorem

I am a new student of Geometric Group theory, and my professor walked us through a proof of the Nielsen-Schreier Theorem that uses the fact that a group that acts freely on a tree must be free. Our ...
112 views

### Permuting subgroups with the same finite index

Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ and $H$ is a subgroup of $G$ with finite index $m$. Let $\phi$ be an automorphism on $G$. ...
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### Writing $G/[G, G]$ as a direct product of cyclic groups

Let $G$ be the group given by the presentation $\langle x , y , z : x^2 , y^3 , (xyz)^4 \rangle$. I would like to write $G/[G , G]$ as a direct product of cyclic groups, where $[G , G]$ is the ...
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### Every free group admits a fixed-point-free involution automorphism

This comes from an exercise in Rotman's "An Introduction to the Theory of Groups": 11.6) Show that a free group $F$ of rank $\geq 2$ has an automorphism $\phi$ with $\phi(\phi(w)) = w$ for ...
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1 vote
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### Finding a group with minimal generators and then a subgroup generated by these as an index two normal subgroup.

Given a group with seven generators and seven relations, each of length 3, how can I use GAP to find the group generated by only three of its generators? For example, G = \langle a,b,c,d,e,f,g \mid ...
• 281
1 vote
52 views

### Defining a map on a subgroup of a free group

Given a set $S$, we write $G(S)$ for the free abelian group on the basis $S$. Given a subset $T\subseteq S$, let $H$ be the subgroup of $G(S)$ generated by $T$. I wonder if the following is true: Can ...
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### Finding special presentations for finite groups

Let $G$ be a finite group. Call a presentation of $G$ "normalised" (I do not know whether such presentations by generators and relations have been studied before and I invented the name &...
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### Finitely presented group with all rank $2$ subgroups not finitely presented?

Is there a finitely presented group $G$ where every noncyclic subgroup $H$ of $G$ that is generated by $2$ elements is not finitely presented? Context: I was wondering about subgroups of finitely ...
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