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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group presentation with circularly shifted relator

Let $G=\langle S \mid R_1 \cup R_2 \cup R_3 \rangle$ be a group presentation with $S=\{a,b,c\}$, $R_1=\{aa{^{\text{-}1}}, bb{^{\text{-}1}}, c^2\}$, $R_2$ the set of all circular shifts of the word $w=...
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2 votes
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Is there an algorithm that can compute finite presentations for finitely presentable subgroups in a FP group with solvable word problem?

Given a group and its finite presentation $G=\langle A\mid R\rangle$, I want the following algorithm: Input: a finite set $W$ of words in $A\cup A^{-1}$ that generates a finitely presentable subgroup ...
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4 votes
1 answer
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Word problem in residually finite groups - enumerating normal subgroups

I am trying to prove that the word problem is solvable for residually finite, finitely presented groups. It is known that one runs two algorithms in parallel. One algorithm stops when the word $w$ is ...
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3 votes
2 answers
159 views

Existence of automorphism taking a word to its inverse

Let $F_2$ be free group on free generators $\{x,y\}$. We know that the inverse of $[x,y] = xyx^{-1}y^{-1}$ is $[y,x]$ and if we take automorphism $\phi$ generated by $\phi(x)=y, \phi(y)=x$ then $\phi$ ...
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2 answers
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If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can the fixed points form a non-normal subgroup?

Suppose that we have a finitely generated and residually finite group $G = K\rtimes\mathbb{Z}$, but $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0)\mid ...
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1 vote
1 answer
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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 ...
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1 vote
0 answers
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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
0 answers
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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 ...
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4 votes
1 answer
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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Finding relators of a matrix group

Let $f_1,\dots,f_n$ be maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $f_i(x) := a_ix + b_i$ with $a_i,b_i \in \mathbb{Q}$. We construct the transformation group $G = \langle f_1, \dots, f_n \...
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1 answer
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Tietze transformations and the trivial group

Suppose you have a finite presentation of a group and you want to determine if it yields the trivial group. We know this is unsolvable in general. But say you start from the trivial group and “...
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2 votes
1 answer
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non finitely generated group with finite abelianization

Suppose $K$ is a non finitely generated and residually finite group, is it possible that $K$ has finite abelianization, i.e. the quotient group $K \big/ [K,K]$ is finite? If we take $K$ to be the ...
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Maximal permutation groups where $1$ is fixed iff $2$ is fixed

Let $S_n$ denote the symmetric group on $\lbrace 1,2,\ldots ,n\rbrace$, for $n\geq 2$, and let $$ \begin{align} X&= \lbrace \sigma \in S_n \mid \sigma(1)=1 \Leftrightarrow \sigma(2)=2\rbrace \\ &...
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subgroup generated by n-th powers of elements in a free group [duplicate]

Let $F=\langle x_1,\ldots,x_n \rangle $ be a finitely-generated free group. For $n \in \mathbb{N} $, let $H_n$ be the subgroup of $F$ defined by: $H_n= \langle x^n|x\in F \rangle$. Is it true that $|F:...
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1 answer
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Looking for a permutation with disjoint image in a generated subgroup

Let $n,m$ be integers with $m\geq 2n, n \geq 1$, and $A=\lbrace 1,2,\ldots, n \rbrace$. Let $S_m$ be the symmetric group on $[|1..m|]$. For every $k\in[|1,n|]$, take a permutation $\sigma_k\in S_m$ ...
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Survey texts on Fox calculus

I am studying the paper by W. Goldman "The symplectic nature of fundmental groups of surfaces" part3, from page 219 where Fox Calculus is used to calculate the Zariski tangent space of $\...
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  • 244
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1 answer
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Generators give quotients of free products of groups?

I have some confusion here. I have this idea that: If $\Gamma$ is generated by (all of) the elements of subgroups $G_1,\dots,G_k$, then $\Gamma=\langle G_1,\dots, G_k\rangle$ is a quotient of the ...
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3 votes
2 answers
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A certain free product of groups is virtually torsion-free

Suppose that $G_1,\ldots,G_n$ are finite groups, and $m\geqslant 0$ is some integer. Set $$G=G_1\ast\cdots\ast G_n*F_m,$$ (where $F_m$ is the free group on $m$ generators). Then, is $G$ virtually ...
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3 votes
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If $A\ast_C B$ is finitely generated, are $A$ and $B$ finitely generated?

I know that if $A$ and $B$ are group of finite rank $n$, there is an amalgamated product $A\ast_{F_2} B$ of rank $2$. It is know that for every $n$ there exist groups $A$ and $B$ of rank $\ge n$ and ...
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Given $H=\langle X_H\mid R_H\rangle$ and $K=\langle X_K\mid R_K\rangle$, find a presentation for Robinson's $H\circ K$

According to this search, this question is new to MSE. The Details: Paraphrasing Robinson's, "A Course in the Theory of Groups (Second Edition)", we have the following Definition: Let $G$ ...
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4 votes
1 answer
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Let $G = \langle x, y \mid x^{7} = y^{3} = e,\; yxy^{−1} = x\rangle$. What is $|G|\,$?

Let $G = \langle x, y \mid x^{7} = y^{3} = e,\; yxy^{−1} = x\rangle$. What is $|G|\,$? What I've done so far: $$yxy^{-1} = x \implies yx = xy.$$ The group $G$ is an abelian group. I can only think ...
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  • 325
1 vote
1 answer
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Every element in a free group is conjugate to a cyclically reduced word

Given a free group $F$ generated by a set $X$, which by definition is the set of reduced words in $X \cup X^{-1}$, with reduced concatenation of words, I've come across a statement that says every ...
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7 votes
0 answers
101 views

Does this group construction preserve finite presentability?

Suppose $G$ is a group. Consider the set $G^G$ of all functions $G \to G$, which forms a group under elementwise multiplication. Now, for all $g \in G$ let’s define $c_g \in G^G$ as the constant ...
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  • 14.8k
1 vote
2 answers
105 views

proof of diamond lemma

I am trying to prove the diamond lemma: Suppose we have two elementary cancellations of a word $w$ then there exists some $w'$ such that there are (possibly trivial) cancellations The diamond lemma, ...
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Chromatic Polynomial- Mobius Inversion Theorem I

can someone here give me a basic approach of Mobius inversion theorem? The theorem goes: Let $N_{e}(x)$ ($N$ sub equal to) be a real-valued function defined for all $x$ in a locally finite partially ...
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4 votes
1 answer
76 views

Center of a quotient of a free group

I encountered the following statement while going through an old result by C.F.Miller (Ref. The Schur Multiplier, Gregory Karpilovsky, Theorem 2.6.6, Page 72): Let $G=F/R$ be a presentation of an ...
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0 answers
84 views

What does $\bigoplus\limits_{\mathbb N} \mathbb Z$ mean?

In this question, $\bigoplus_{\mathbb N} \mathbb Z$ is given as an example of a group that is freely generated but not finitely generated. A similar question is asked here but it has been closed ...
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9 votes
1 answer
247 views

Prefixes of a word multiplying to the identity in a free group

Let $A$ be a finite alphabet, and let $w \in (A \cup A^{-1})^\ast$ be a freely reduced word over the alphabet $A$ and formal inverse symbols $A^{-1}$. Suppose $w$ is non-empty. Can there ever be non-...
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2 votes
1 answer
118 views

Does the Baumslag Solitar group $B(2,3)$ contain a non-trivial element with arbitrary roots?

The Baumslag Solitar groups $B(n,m)$ are defined via the presentation $\langle a,b \mid b a^m b^{-1} = a^n \rangle$. We say that an element $g$ of a group $G$ has an $n$-th root, if the equation $g = ...
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1 vote
1 answer
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Why is a free factor of a group malnormal in that group?

For a subgroup $K \leq H$, we say $K$ is a free factor of $H$ if $H$ can be written as the free product $K * C$ for some $C \leq H$, i.e. if we have the presentations $K = \langle S_K \mid R_K \rangle$...
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  • 193
1 vote
2 answers
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Showing $\mathbb Z_6\cong \langle x,y\mid x^3=1,y^2=1,xyx^{-1}y^{-1}=1\rangle.$

Let $$B:=\langle x,y\mid x^3=1,y^2=1,xyx^{-1}y^{-1}=1\rangle$$ My question is: how does one show that $$\mathbb Z_6 \cong B?$$ My solution: \begin{array}{ccc} y & \mapsto & 3\\ x & \mapsto ...
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  • 1,561
1 vote
1 answer
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Endomorphism monoid of free group

The automorphism group of $F_2$(the free group generated by two elements) is finitely generated. Especially, it is generated by Nielsen transformations. If we consider all homomorphisms from $F_2$ to ...
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2 votes
1 answer
87 views

Presentation of a group with the same number of generators and relations

This may be a silly question as I do not know much about presentations of groups. Let $\langle S\mid R\rangle $ be a finite presentation of a group. Is it always possible to get a finite presentation $...
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0 votes
0 answers
37 views

How can I create the presentation (min. set of relations) of the quaternion group of order 8? How can I look for conjugation relations?

What are the steps that leads me to know the presentation (specifically defining the generators and the minimum set of relations required to define this group)of the quaternion group of order 8? What ...
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1 vote
0 answers
45 views

Using Kurosh’s Theorem to study elements of $PSL(2, \mathbb{Z})$

Ok, I believe this is a relatively basic question, but I couldn’t figure out what I’m doing wrong. What I’d like to prove is the following result: If an element $T$ in $SL(2, \mathbb{Z})$ has finite ...
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  • 1,497
4 votes
1 answer
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The operation rule of the semidirect product $C_m\rtimes C_k$

It is known that the semidirect product $C_m\rtimes C_k$ is defined by presentation $$ C_m\rtimes C_k=\langle a, b\mid a^m=1, b^k=1, b^{-1}ab=a^e\rangle, $$ where $e^k\equiv1\pmod{m}$. Since every ...
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6 votes
1 answer
79 views

Relations from quotient of free product

This question arose from an exercise which asks you show that the fiber coproduct exists in the category of groups. I was eventually able to (mostly) solve the problem by “gluing” the images of ...
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4 votes
1 answer
60 views

Shortest equivalent word for FPG

Given a finitely presented group $G$ and a word $w$ in $G$, is there always an efficient algorithm to find the set of shortest words equivalent to $w$? I am guessing that the problem of finding the ...
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2 votes
1 answer
79 views

Presentation for Binary Icosahedral Group "Using" Presentation for $A_5$

This is probably a stupid question, but, per this post Group presentation of $A_5$ with two generators, a presentation for $A_5$ is given by $A_5 \cong \langle x,y \mid x^5=y^2=(xy)^3=1 \rangle$. In ...
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0 votes
0 answers
66 views

Freiheitssatz and Word Problem; a problem with literature terminology

One of the classical achievements of the combinatorial group theory is the decidability of the word problem in a finitely generated group with one defining relation. This result was a corollary of a ...
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  • 1,188
1 vote
3 answers
90 views

What can the order of the generators of a non abelian group tell us about the group?

What can the order of the generators of a non abelian group tell us about the group? For example, if we have $G$ as the non abelian group of order $8$, representing the quaternions; $$ G=\{s,t\mid s^...
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3 votes
0 answers
32 views

Is there a systematic way to find the center of a finitely-presented, finite $p$ group?

Given a finitely-presented and finite $p$-group $G = \langle S \mid R \rangle$, is there a systematic way to find its center? My end goal is to find a subnormal chain of every order $1, p, p^2, \dots, ...
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  • 4,468
7 votes
2 answers
132 views

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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  • 3,266
2 votes
2 answers
104 views

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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  • 1,497
1 vote
1 answer
68 views

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 ...
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  • 281
1 vote
1 answer
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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2 votes
0 answers
74 views

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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  • 2,030
3 votes
0 answers
114 views

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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  • 5,053
3 votes
1 answer
65 views

Show finite $\langle x,y,z,t\rangle$ with relations $y^x=y^2$, $z^y=z^2$, $t^z=t^2$, $x^t=x^2$ is trivial.

This is Exercise 3.2.8 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search on Approach0, it is new to MSE, although this similar question that is in ...
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