Questions tagged [group-presentation]

For questions concerning groups defined via a presentation by generators and relations.

30
votes
6answers
1k views

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 ...
23
votes
1answer
3k views

Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
18
votes
1answer
445 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
18
votes
0answers
210 views

Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?

I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.) Let $G$ be a group, and let $a,b,c\...
16
votes
2answers
1k views

Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group. I have tried all ...
15
votes
8answers
2k views

Why the group $\langle x,y\mid x^2=y^2\rangle $ is not free?

Why is the group $G= \langle x,y\mid x^2=y^2\rangle $ not free? I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
14
votes
3answers
2k views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
12
votes
3answers
2k views

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 ...
12
votes
3answers
337 views

Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.

I'm trying to prove that the following group has infinite order: $$H=\langle a,b\mid a^{3}=b^{3}=(ab)^{3}=1\rangle.$$ Currently I'm checking on some cases using the relations, but my problem is the ...
12
votes
0answers
1k views

Finitely generated group which is not finitely presented [duplicate]

Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented? I was looking for a finitely presented group with a central ...
12
votes
0answers
186 views

Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with $2$ generators? More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of ...
10
votes
1answer
196 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. ...
9
votes
3answers
388 views

Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?

All is in the title: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$? I think it is the case, but I don't know how to prove it.
9
votes
1answer
2k views

Intuitive understanding of the Reidemeister-Schreier Theorem

I am reading Combinatorial Group Theory by Lyndon and Schupp, and I'm having some trouble getting through the proof of the Reidemeister-Schreier theorem (page 103 in that book) - you can read that ...
9
votes
0answers
140 views

Is this specific group finite?

I have the following group presentation: $G=\left\langle a,b,c\ |\ a^2,b^{11},(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},c^2,(ac)^3,(bc)^2\right\rangle$ Is $G$ finite? GAP's Size(G) runs ...
8
votes
1answer
545 views

Is this a group? If so, what group is it?

I have the following group (at least, I think it's a group) generated by $\langle a,b,c \rangle$ where the operation $\cdot$ obeys the following rules: $a^2=b^2=c^2=1$ (where $1$ is the identity). $\...
8
votes
1answer
627 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take $G=...
8
votes
1answer
408 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that $\...
8
votes
0answers
196 views

Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
8
votes
0answers
581 views

Way to Tietze's Transformation Theorem

During our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
7
votes
1answer
287 views

One relator groups shortest word in quotient

Suppose we have a group $\langle S \vert r \rangle \cong F_S /\langle \langle r \rangle \rangle$ where $S$ is a finite set of generators and $r \in F_{S}$, i.e. a finitely generated one realtor group. ...
7
votes
1answer
207 views

On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. EDIT In fact, these are ...
7
votes
1answer
337 views

Finitely Presented is Preserved by Extension

Given $N= \langle n_i|r_j \rangle$ and $G/N= \langle g_k|s_l \rangle$, how do we prove $G$ has a finite presentation? We know that $G$ is f.g. by $\{n_i,g_k\}$ (I am being sloppy about directly ...
7
votes
1answer
274 views

Does this group presentation define a nontrivial group?

Given a presentation $$ \langle x,y,z : x^y=x^2, y^z=y^2, z^x = z^2 \rangle, $$ where $x^y$ is just the usual conjugation. Can we say for sure, whether this presentation defines a nontrivial group?
7
votes
2answers
150 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
7
votes
1answer
79 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
7
votes
2answers
564 views

Group presentations: What's in the kernel of $\phi$?

I have a question about group presentations (in terms of generators and relations). It's been really bugging me for ages. Would really appreciate any thoughts on this. Cheers, Michael You are 'given' ...
7
votes
2answers
1k views

Presentations for alternating groups

Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators $a_n=(123),b_n=(1,2,\ldots,n)$?
6
votes
4answers
2k views

Does a Conjugacy Class always contain an element and its inverse?

The definition of a conjugate element We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$ Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = \big<a,...
6
votes
2answers
598 views

notations of generators and relations

I need to understand the following about generators and relations notations: Is $\langle a,b \mid a^kb^l\rangle =\langle a,b\mid a^k=b^l\rangle =\langle a,b\mid a^k,b^l\rangle$? Is $ \langle a,b\mid ...
6
votes
3answers
330 views

What is this group? (Recognising a group from a presentation).

I am trying to find out what the following group is: $$G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3\rangle.$$ Due to the isomorphism problem for groups, there is no algorithmic way to approach ...
6
votes
2answers
112 views

Presentation of Groups

I have troubles to solve this kind of exercises. For example: Let $$G_1=\langle x,y |x^3=y^4=1\rangle,~~~G_2=\langle x,y |x^6=y^6=(xy)^3=1\rangle. $$ I want to check that $G_1$ is an infinite ...
6
votes
1answer
91 views

Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
6
votes
1answer
73 views

Representation of elements by words on finite groups.

Consider a finite group $G$ of order $n$, that is generated by two elements $a,b$. I would like to find some good upper bound $m$, such that for any element $g \in G$ there is an expression of $g$ ...
6
votes
1answer
195 views

Is the minimum number of relations in a free product, the sum of the minimum number of relations in the free factors?

Say $\rho(G)$ is the minimum number of relations required to present the group $G$. Is $\rho(A*B)= \rho(A)+\rho(B)$? What can be said about $\rho(A*B)$? A while ago I was thinking about $C_3*C_4$, ...
6
votes
1answer
617 views

The order of a group presentation

Find the order of the group $G$ which has the presentation $\langle a,b \mid a^{16}=b^6=1,bab^{-1}=a^3\rangle $ I found that $a^8b=ba^8$ hence $\langle a^8,b\rangle$ is an abelian sungroup of $G$. ...
6
votes
4answers
1k views

free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$

I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication (...
6
votes
1answer
157 views

on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
6
votes
2answers
144 views

Explicit description for $G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle$

I am trying to give an explicit description of the group $$G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle\,.$$ Generalizing to fewer generators, one ends up with the trivial group, i.e. $$...
6
votes
0answers
155 views

Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
5
votes
6answers
367 views

A presentation of a group of order 12

Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$. I tried to let $d=ab\Rightarrow G=\langle d,c\mid d^2 =...
5
votes
2answers
303 views

Showing that a group with a presentation is free/not free

Show that the group with presentation $\langle a, b, c \mid a^2cb^3\rangle$ is free with basis $\{a, b \}$. Show that the group with presentation $\langle a, b, c \mid a^3b^3 \rangle$ is not free. I'...
5
votes
2answers
633 views

Understanding group presentation as a quotient

I'm just starting to learn a little group theory, so please forgive any ignorance I demonstrate in the following. I'm trying to understand the concept of a group being defined based on its ...
5
votes
2answers
563 views

Is there software to help with group presentation

I wrote a computer program that generates group presentations. I would like to know the sizes of the resulting groups. I know that this is undecidable. Are there good heuristic programs that can ...
5
votes
2answers
153 views

Efficient computation of conjugacy classes of a small group.

I'm trying to construct a character table for a group of order 54 given by: $$ \langle a,b : a^9 = b^6 = 1, b^{-1} a b = a^2\rangle $$ To do this first I need to compute conjugacy classes. This ...
5
votes
2answers
141 views

What is this group $G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle$

Consider the group presentation $$G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle.$$ Is this a known group? What is $G$ isomorphic to? Thanks a lot.
5
votes
2answers
64 views

Describing $\langle x,y : x^{2} , y^{3} , [x,y] , x^{6}y^{6}\rangle$.

I am trying to identify the following presentation $\langle x,y : x^{2} , y^{3} , [x,y] , x^{6}y^{6}\rangle$ I substituted the first relation in the final one and got $x^{6}=1$ so the group is ...
5
votes
3answers
297 views

What is needed to specify a group?

I have come across several groups, some of which have the same number of generating elements and of the same orders. Take, for instance, $D_{2n}$ and $S_n$. I have never seen it read explicitly, but ...
5
votes
2answers
198 views

How to obtain a presentation for each group of order $64$

I am new to his forum, and would like to know how to obtain a presentation for each group of order $64$. I wish to do this for all the groups of order $64$. Thanks in advance.
5
votes
1answer
110 views

How to prove these two groups are isomorphic

If $G_{1}$ is $\langle a,b \mid a^2 = b^2 \rangle$ and $G_{2}$ is $\langle p,q \mid pqp^{-1} = q^{-1} \rangle$, find an isomorphism $\phi : G_{1} \rightarrow G_{2}$. I tried the obvious by letting $a$...