Questions tagged [group-presentation]

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

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18
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 ...
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 ...
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 (...
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 ...
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 ...
5
votes
1answer
2k views

Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an $...
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 ...
3
votes
1answer
1k views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by $$D_n=\{I,...
2
votes
0answers
281 views

Group Presentation of the Direct Product.

This is Exercise 1.2.5 and Exercise 1.2.6 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Details: Definition 1: Let $$\langle a,...
2
votes
2answers
626 views

Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? [duplicate]

Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? I've just been playing around with the relations but I can't seem to get ...
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.
7
votes
1answer
280 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?
12
votes
3answers
339 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 ...
4
votes
2answers
514 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
7
votes
1answer
343 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 ...
6
votes
1answer
627 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 $G$. ...
2
votes
2answers
884 views

Generalized Quaternion Group

Let $w = e^{\Large\frac{i\pi}{n}} \in \mathbb{C}.$ Prove that the matrices $X=\left( \begin{array}{cc} w & 0 \\ 0 & \overline{w} \\ \end{array} \right)$ and $Y = \left( \begin{...
2
votes
2answers
164 views

Identifying $\langle a,b\mid a^2b^2\rangle$.

The Question: What group is $G=\langle a,b\mid a^2b^2\rangle$? Thoughts: I found that the presentation maps onto $\langle a, b\mid a^2, b^2, 1\cdot 1\rangle\cong \mathbb{Z}_2\ast\mathbb{Z}_2$, ...
7
votes
2answers
567 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' ...
3
votes
1answer
385 views

Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$.

I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look ...
1
vote
0answers
99 views

Cyclically Presented Groups.

I suppose I've answered my own question in writing it but some input would be great. The Question(s) What exactly are cyclically presented groups, what are some examples, and where might I find ...
1
vote
1answer
620 views

Commutator subgroup of a finitely generated nilpotent group.

Let $G$ be a finitely generated nilpotent group. Is the commutator subgroup $[G,G]$ finitely presented? Edit: I am also interested in the weaker question: is $[G,G]$ finitely generated?
1
vote
1answer
279 views

Group presentation for discrete Heisenberg group

I have to show that the discrete Heisenberg group H (with $a,b,c \in \mathbb{Z}$) has the presentation $H=\langle x,y\mid [[x,y],x]=[[x,y],y]=1 \rangle$. I figured we write $H$ as a group of triples $...
0
votes
0answers
187 views

groups of order $p^4$

I need the classification of finite non-abelian groups of order $p^4$ from E. Schenkman's book "Group theory, 1965". Unfortunately our library has no this book and there does not exist the full ...
10
votes
1answer
201 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
1answer
633 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=...
5
votes
0answers
258 views

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$… [closed]

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...). It's ...
5
votes
6answers
368 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 =...
4
votes
2answers
935 views

If I have the presentation of a group, how can I find the commutator subgroup of it?

I have the group given by the presentation $G= \langle a,b\mid a^2,b^2\rangle$ How can I in general find $G',G/G',G''$ ? thanks for any hints.
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.
6
votes
0answers
159 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
0answers
258 views

Presentations of Semidirect Product of Groups

I have seen here that given two groups $G=\langle X|R \rangle := F(X)/N(R)$ and $H=\langle Y|S \rangle := F(Y)/N(S)$, then their semidirect product can be written as: $$ G\rtimes_\phi H \;=\; \langle ...
5
votes
2answers
638 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 ...
4
votes
3answers
107 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
votes
1answer
178 views

How to show this presentation of the additive group $(\mathbb{Q},+)$?

The task is: Show that $$ \langle (x_n)_{ n \in \mathbb{N}} \mid x_n^n = x_{n-1} \text{ for } 1 < n \in \mathbb{N} \rangle $$ is presentation of additive group $(\mathbb{Q},+)$. Can you explain ...
3
votes
1answer
787 views

Presentation of abelian group

How one can find the abelian group which has a presentation $$\langle x,y,z,w\mid6x+8y+10z+14w, 4x+4y+4z+4w\rangle$$ Is there any way indicates the steps to find such a group? Or just by guesswork ...
2
votes
0answers
765 views

Proof of every finite group is finitely presented.

I'm reading the proof that every finite group is finitely presented from Dummit's Abstract Algebra, but there's a part that I don't understand. In the proof below, what are the elements $\tilde{g_i}$? ...
2
votes
0answers
58 views

Finding $m,n,k$ for which $|G|=mn$ and $G$ is nonabelian, where $G$ is given by $\langle x,y\mid x^m=y^n=1,xy=yx^k\rangle$.

$G$ is defined by the relations $x^m=y^n=1,xy=yx^k$. For which $m,n,k$ does this give a nonabelian group order $mn$? I started playing around with this using GAP, starting with the case where $m=13$....
1
vote
2answers
179 views

Defining Groups via the Universal Property

The snippets below on group presentation led to the following questions: What is the explanation/intuition of the universal property. Basically what (2) is saying. I understand free groups (though ...
1
vote
0answers
220 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
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 ...
5
votes
0answers
148 views

How can we determine the group with the presentation $G:=\left\langle g,h\mid g^4=h^4=1,hg=g^{-1}h \right\rangle?$

Please how can we determine the finite group whose presentation is given as: $$G:=\left\langle g,h\mid g^4=h^4=1,hg=g^{-1}h \right\rangle?$$ See this comment for some context from the person asking ...
5
votes
0answers
294 views

Has every finite group a minimal presentation?

For a finite group $G$ let $d(G)$ be the minimal number of generators of $G$ and let $r(G)$ be the minimal number such that $G$ has a finite presentation with $r(G)$ relators. Call a presentation with ...
3
votes
1answer
733 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of "...
2
votes
1answer
201 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi H \;...
2
votes
1answer
1k views

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can't find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the ...
1
vote
1answer
768 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
1
vote
1answer
55 views

Finding the index of $N$ in $F$.

This is Exercise 2.1.4(a) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
0
votes
2answers
53 views

Identifying $\langle x_e, x_a, x_b, x_{ab}\mid x_ex_a=x_{ab}, x_ax_e=x_b, x_bx_{ab}=x_a, x_{ab}x_b=x_e\rangle.$

Consider the group presentation $$\langle x_e, x_a, x_b, x_{ab}\mid x_ex_a\stackrel{(1)}=x_{ab}, x_ax_e\stackrel{(2)}=x_b, x_bx_{ab}\stackrel{(3)}=x_a, x_{ab}x_b\stackrel{(4)}=x_e\rangle.$$ What group ...
-1
votes
2answers
95 views

Show that the class of the word $aba$ in group $\langle{a,b\,\vert\,a^{2}b^{2} }\rangle$ is not the trivial element. [closed]

Could anyone give me a suggestion to solve this problem about group presentations? Show that the class of the word $aba$ in group $\langle{a,b\mid a^{2}b^{2} }\rangle$ is not the trivial element.