Questions tagged [combinatorial-group-theory]

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

Combinatorial group theory deals with free groups and presentations of a group by generators and relations. Combinatorial group theory is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. Combinatorial group theory is largely subsumed by geometric group theory.

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subgroups of finitely generated groups with a finite index

Let $G$ be a finitely generated group and $H$ a subgroup of $G$. If the index of $H$ in $G$ is finite, show that $H$ is also finitely generated.
Nana
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How do you prove that a group specified by a presentation is infinite?

The group: $$ G = \left\langle x, y \; \left| \; x^2 = y^3 = (xy)^7 = 1\right. \right\rangle $$ is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a finitely presented group, I could do a coset…
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Subgroups defined by negative formulas

I start with a simple problem that I was able to solve: Let $G$ be a group. Let $a\in G$. Assume that $H := \{g \in G : g^2 \neq a\}$ is a subgroup of $G$. The question: Can we define $H$ with a "positive" formula, not involving the symbol $\neq$?…
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Can the map sending a presentation to its group be considered as a functor?

It is well-known that the functor $Grp \to Set$ sending a group $G$ to its underlying set $UG$ has a left adjoint, the functor $Set\to Grp$ sending a set $X$ to the free group $FX$. I wonder whether one can consider the following modification of the…
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Probability of a group being finite

Suppose $F_m := F[x_1, … , x_m]$ is a free group on $m$ generators $x_1, … , x_m$ and lets define Cayley ball $B_m^n := \{e, x_1, x_1^{-1}, … , x_m, x_m^{-1}\}^n$ as the set of all elements with Cayley length $n$ or less. Suppose $R_1, … , R_l$ are…
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Homomorphisms of $\mathbb{F}_2$ that preserve $aba^{-1}b^{-1}$

Let $\mathbb{F}_2$ be the free group generated by $a$ and $b$. Suppose we are given a homomorphism $\phi: \mathbb{F}_2 \to \mathbb{F}_2$ with the property that $\phi(aba^{-1}b^{-1}) = aba^{-1}b^{-1}$. Can I conclude that $\phi$ is surjective? Can…
user101010
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When is a group isomorphic to the infinite cyclic group?

I am learning algebra and I am a bit confused. Let's say I have a finitely presented group $G$, can anyone tell me if it is possible to find out if $G\cong \mathbb{Z}$? Thanks
13
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The word problem for finite groups

The word problem for finite groups is decidable. Is it obvious that this is true? In particular, I'm not entirely sure about what it means for the problem to be decidable (in this case---I think I understand what decidable means in general). I…
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Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to finish the following question (I am ok with general…
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Are all almost virtually free groups word hyperbolic?

Suppose $G$ is a finitely generated group with a finite symmetric generating set $A$. Lets define Cayley ball $B_A^n := (A \cup \{e\})^n$ as the set of all elements with Cayley length (in respect to $A$) $n$ or less. Suppose $R_1, … , R_k$ are $k$…
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Can we apply relations in a group presentation one by one?

Consider a group presentation $\langle x,y \mid xy=yx, x^7=y^3 \rangle$. By definition, this is $F(\{x,y \})/N(xyx^{-1}y^{-1},x^7y^{-3})$ where $F(S)$ denotes the free group on the set $S$ and $N(R)$ denotes the normal subgroup generated by $R$.…
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What is the abelianization of $\langle x,y,z\mid x^2=y^2z^2\rangle?$

Let $G=\langle x,y,z\mid x^2=y^2z^2\rangle$. What is the abelianization of this group? (Also, is there a general method to calculate such abelianizations?) Update: I know how to get a presentation of the abelianization by adding relations like…
10
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Is a HNN extension of a virtually torsion-free group virtually torsion-free?

Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free group. Let $H,K
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Probability and Combinatorial Group Theory.

If this is too broad or is otherwise a poor question, I apologise. I learnt recently that the probability that two integers generate the additive group of integers is $\frac{6}{\pi^2}$. What other results are there like this? I'm looking for any…
Shaun
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Presentations of subgroups of groups given by presentations

If you have a finitely generated group given by a presentation, is there a good method to determine a presentation for a subgroup generated by some subset of the generators given in the presentation? Specifically: I have found a presentation for the…
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