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Questions tagged [combinatorial-group-theory]

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

2
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3answers
46 views

Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
2
votes
1answer
36 views

Reference Request for Small Cancellation Theory

I am looking for a self contained survey / paper / lecture notes on small cancellation theory and it's generalizations. I am aware of Lyndon and Schupp's textbook chapter and I have been recommended ...
1
vote
1answer
54 views

Prove or disprove that $G = \langle{x,y\;|\; x^3, y^3, x^{13}, [x,y]=1}\rangle$ is trivial.

Prove or disprove that $G = \langle x,y \mid [x,y]=x^3=y^3=x^{13}=1 \rangle$ is trivial. So the fact that $x^3=x^{13}$ means that the order of $x$ divides both $3$ and $13$, thus $|x|=1$ and so x is ...
2
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1answer
47 views

Understanding HNN extensions: intuition, examples, exercises.

What is an HNN extension? What would be some elementary, intuitive examples of them and what exercises involving them would you suggest? The Wikipedia definition is easiest to get to, since neither ...
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0answers
88 views

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 $H,K<G$ be isomorphic (finite index) subgroups of $G$ and let $\varphi:H\rightarrow K$ be an isomorphism. ...
1
vote
1answer
41 views

Example III.3.2(1) of Baumslag's “Topics in Combinatorial Group Theory”: proving $F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$ is free.

This question is a little tricky (for me, at least), since in the textbook the proof of Theorem 5: Every subgroup of a free group is free. is not yet provided (even though I've seen such proofs ...
3
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0answers
23 views

Is there a formula for $[F_n : V_{\{x^3\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_3(n) = [F_n : V_{\{x^3\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_3(n)$? ...
3
votes
1answer
55 views

Finding the order of $\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle.$

Im new at abstract algebra stuff and im wondering whats the technique to prove this kind of stuff. Question: Let $G=\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle$, prove that $|G|=16 $ and find all ...
2
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1answer
29 views

Are upper quasiverbal and lower quasiverbal subgroups always the same subgroup?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a prevariety described by $Q$ as a class ...
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1answer
50 views

$\pi_1(M)=\langle a,b|b^{-1} b^{-1},a \rangle \cong \Bbb Z?$ [closed]

$\pi_1(M)=\langle a,b|b^{-1} b^{-1},a \rangle \cong \Bbb Z?$ Why? Thanks in advance!
2
votes
3answers
88 views

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

I'm trying to prove that $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$. If they are isomorphic, then their abelianizations $G/[G, G] = \...
1
vote
1answer
28 views

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

Let $G$ be a one-ended hyperbolic group. Can $G$ contain an $\infty$-ended group? If it can, are there any conditions on $G$ beyond hyperbolic which makes it impossible? As a particular example, if $...
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0answers
67 views

Reduced word is uniquely written

Given a family of groups $\{G_i:i\in I\}$ we may assume that $G_i$ are pairwise disjoint. Let $X=\bigcup^{}_{i\in I}G_i$ and let $\{1\}$ be one element disjoint from $X.$ A word on on $X$ is any ...
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0answers
25 views

Symmetrising the relations in a presentation of a group

Let $G$ be a finitely presented groups defined by $$G=\{x_1,\ldots,x_n\mid R_1(x_1,\ldots,x_n)=\cdots=R_m(x_1,\ldots,x_n)=1 \}.$$ Let this presentation be denoted by $P$. Let $S_n$ be the ...
2
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1answer
82 views

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

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
3
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1answer
37 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?

Does exist a finite group $G$ and fixed $k \geq 1$ such that the $n$-direct product $G^n = G \times \dots \times G$ is $k$-generated for every $n \geq 1$? I suspect the answer is no. Does exist a ...
2
votes
1answer
60 views

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

I wonder whether the following statement is true: Let F is a group and X is a subset of F. Then $\left< X \right> =F\quad \Longleftrightarrow \quad$ For any group G and any function $\phi :X\...
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1answer
52 views

Fixed points of a group action on tree

Suppose a group $H$ acts on a tree $T$, and this action fixes a point. Let $T_1$ be an $H$ invariant subtree of $T$. How do I show that $H$ fixes a point in $T_1$?
3
votes
3answers
92 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?

If we are handed the group 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? Nothing in this presentation tells us that $i^2=...
3
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1answer
50 views

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

Let $G$ be a group and $H$ a subgroup of the center of the group $G$ (so $H$ is normal in $G$). Suppose that $G/H$ is a residually finite group, then is it true that $G$ is a residually finite group? ...
3
votes
1answer
55 views

Reference for a result in group theory

There is a question on mathoverflow regarding the finitely generated center of a finitely generated group. In the first remark of the question it is written that "If $G$ is a finitely generated ...
3
votes
1answer
45 views

Proving the Free Abelian Group is Free Abelian…?

On page 40 of these notes is the following exercise: Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$. On ...
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vote
2answers
34 views

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

Say $X$ is a finite set, $F \langle X \rangle$ is the free group on the set $X$ and $G$ be a group. If I have a surjective homomorphism $$\varphi : F \langle X \rangle\longrightarrow G$$ then can I ...
3
votes
1answer
70 views

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

Suppose we are given the following presentation of the quaternion group: $Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$ Is it obvious that $i^4 = 1$?
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votes
3answers
181 views

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

Do there exist such nontrivial groups $A$, $B$, $C$ and $D$, such that $A \times B \cong C \ast D$? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction....
2
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1answer
48 views

What is an algorithm for determining if a finitely presented group is finite

Suppose I am given a presentation of a group with a finite number of generators and a finite number of relations. Is there an algorithm for determining if the group is finite? Also, if there is such ...
4
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1answer
48 views

Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?
1
vote
1answer
67 views

A proof that $\langle u,v\mid u^4=v^3=1, uv=v^2u^2\rangle$ defines the trivial group.

This appears to be new to MSE. I'm reading "Abstract Algebra (Third Edition)," by Dummit & Foote. This is based on Exercise 1.2.18. Question: Show that $$Y=\langle u,v\mid u^4=v^3=1, uv=v^2u^...
1
vote
1answer
53 views

Derived subgroup of $\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$. [closed]

I would like to prove that if $M_n(p)=\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$, then $M'_n(p)$ is a cyclic group of order $p$. I was wondering if someone could ...
3
votes
2answers
107 views

Why study “virtual properties”?

In group theory, I have seen some results which involve "virtual properties". E. g. virtually abelian, virtually solvable etc. The definition is, according to Wikipedia (https://en.wikipedia.org/wiki/...
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0answers
40 views

Identifying a group involving transformation of 4 integers

I'm looking at some messy data, and I found by accident that some properties seem preserved under some transformations of a subset of the variables. After some work I think I found a base set of ...
6
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2answers
83 views

Free groups are residually of rank 2

Let $F_X$ denote the free group on the set $X$, and $F_n$ the free group of rank $n$. I have read that any free group is residually $F_2$, and I was trying to understand this. For any free group $F$,...
4
votes
2answers
128 views

How to show that $\langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle$ is not Abelian?

I'd like to show that $$ G = \langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle $$ is non-Abelian. I have tried finding a surjective homomorphism from $G$ to a non-Abelian group, but I haven't found one....
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0answers
68 views

What is a presentation of the upper triangular subgroup of $GL(2, \mathbb Q)$?

I have been trying to find a presentation of the upper triangular subgroup of $GL(2, \mathbb Q)$ by considering the free group $Fr(\{x_i| i\in \mathbb Q\})$ under a homomorphism $f$ into $GL(2, \...
2
votes
1answer
79 views

Kropholler's Type of Tits Alternative for Generalised Baumslag-Solitar Groups.

In "Recent Results on Generalized Baumslag-Solitar Groups," by D.J.S Robinson in Note Mat. 30 (2010) suppl. n. 1, 37–53, it is claimed that Kropholler [ . . . ] showed that there is a type of Tits ...
2
votes
1answer
74 views

The (un)decidability of the Tits Alternative for any given (suitably defined) set of groups.

Please forgive me if this question is ill-formed. I don't know much about decidability. Some Background: There are problems in combinatorial group theory that are undecidable, such as the word ...
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0answers
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Is there an English translation of the German book “Einführung in die Kombinatorische und die Geometrische Gruppentheorie”?

In a recent publication called "Secure passwords using combinatorial group theory," by G. Baumslag et al., it says that the following book is a "standard reference" in combinatorial group theory. ...
3
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1answer
38 views

Criterion for a Finitely Presented Group

Let $G$ be a finitely generated group with a normal subgroup $H$, such that $H$ and the quotient $G/H$ are finitely presented. Does it follow that $G$ is finitely presented? I'm attempting to ...
2
votes
2answers
70 views

Rubiks Cube function — how many configurations reachable in n moves.

I'm working on a relatively low-level math project, and for one part of it I need to find to a function that returns how many many configurations are reachable within n moves. from the solved state. ...
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0answers
68 views

Two presentations of a group, one certainly finite. Need the other be?

I know the answer to the question above is "no", quite flatly. The counter example is below: $$\mathbb{Z}\cong\langle a,b\mid b^2a^{-1}\rangle\cong \langle a,b\mid\lbrace b^{2^{n+1 }}a^{-2^n}:n\in\...
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votes
1answer
75 views

Computing $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$

I want to compute $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$ with respect to homomorphisms: $\varphi_1:\mathbb{Z} \ni n \longmapsto an \in \mathbb{Z}$ $\varphi_2:\mathbb{Z} \ni n \longmapsto bn \in \...
4
votes
3answers
114 views

$G=\langle a,b \mid baba^{-1}=1\rangle$ Show that $\langle a \rangle$ is infinite

Let $G=\langle a,b \mid baba^{-1}=1\rangle$. Show that the subgroup generated by $a$ is infinite. My attempt Suppose $\langle a\rangle$ is finite so $a^k = 1$ for some $k \in \mathbb{Z}$. So I ...
3
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0answers
54 views

on the order of derived subgroup

Let $P=M_{p^{n+1}}=\langle a,b \mid a^{p^{n}}=b^p=1, a^b=a^{1+p^{n-1}}\rangle$. I want to prove that $P^{\prime}=\langle [a,b] \rangle$. My Try: Clearly $\langle [a,b] \rangle \le P^{\prime}$. Put $...
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0answers
53 views

How to prove that two groups with different presentations are isomorphic in a naive way?

One can define a presentation of a group naively (ala Dummit-Foote in Chapter 1.2), i.e., as a group generated by certain elements with certain relations such that all other relations follow from the ...
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3answers
253 views

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 ...
2
votes
3answers
100 views

Show that the group $G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$ has order $27$.

This is Exercise 1.2.21 of Magnus et al's book on combinatorial group theory. The Question: Show that the group $$G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$$ has order $27$...
2
votes
1answer
101 views

Proving $U=U$ is derivable from any set of relators.

This is Exercise 1.1.1 of Magnus et al's book on combinatorial group theory. The details: Let $G=\langle X\mid R\rangle$ for sets $X, R$. Definition 1: The empty word $\varepsilon$ and the words $...
3
votes
1answer
37 views

Cancellation in a presentation of a group

In a method for presenting a subgroup we are given A group $$\ G= \bigl\langle\, x, \ y \mid x^2 yxy^3 , \ y^2 xyx^3\,\bigr\rangle$$ So $$\ G/G' = \bigl\langle\, x, \ y \mid x^2 yxy^3 , \ y^2 xyx^...
1
vote
1answer
28 views

Smith form for a matrix

I have a problem in progressing in SNF We have matrix $A$: $$A= \begin{bmatrix} 14&3&11\\8&-3&11\\3&3&0 \end{bmatrix} $$ I tried to make the first row zeros except the ...
3
votes
1answer
73 views

Terminology: The group(s) of symmetries of the Cayley graph of a group.

Please forgive me if this question is ill-defined. It's late here and I want to ask the question whilst it's still fresh in my mind. Motivation: Suppose we have a group $G$ given by a presentation $$...