Questions tagged [group-presentation]

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

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5
votes
3answers
241 views

Does there exist a finite group with the following presentation?

Let $G$ be a finite group (with only two generators and $m=n$) presented as $$ G = \langle a, b : a^m = b^n = (W(a,b))^p= \ldots\text{other-such-relations}\ldots= 1 \rangle $$ where $m,n,p>1$ , ...
5
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3answers
974 views

Understanding presentations of groups

I'm trying to a build a better understanding of presentations. I get that a a group has a presentation $\langle S \mid R \rangle$ if it is the "freest" group subject to the relations $R$. But, for ...
5
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1answer
743 views

Group theory book: presentations and group actions

I have some basic abstract algebra knowledge (the usual groups/rings/fields). Now I would like to study, in depth, presentations of groups and group actions. (either of which I have no knowledge) ...
5
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2answers
344 views

Fundamental group of Poincaré sphere

Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group? ...
5
votes
2answers
96 views

$G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free.

I have to prove that $G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free. Some things about this group that I understand are first show that $M=\langle (xy)^2,x^2,y^2 \...
5
votes
1answer
47 views

Showing formally that $H:=\langle x,y| x^2, y^n, yxyx^{-1} \rangle$ is a presentation of $D_{2n}$

I want to Show formally that $H:=\langle x,y| x^2, y^n, yxyx^{-1} \rangle$ is a presentation of $D_{2n}$. To start with, by the universal property of the free group, there is a group homomorphism $\...
5
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1answer
309 views

Presentation of a group isomorphic to $A_4$

I have a group $G$ defined by $G = \langle x,y,z|x^2 = y^3 = z^3 = xyz \rangle$ and we know that $a$ $=$ $xyz$ belongs to the centre of $G$. But im struggling to show that $\frac{G}{\langle a\rangle} \...
5
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2answers
94 views

Quotient of group given by presentation is finite

Consider the group $$G = \langle a,b,c ~ \mid ~ a^2 = b^3 = c^5 = abc \rangle$$ Prove that $\mathbf{(a)}$ $abc$ is an element of the center of $G$; and $\mathbf{(b)}$ $G/ \langle abc \rangle$ is a ...
5
votes
1answer
220 views

If $G/N$ and $N$ are finitely presented, then $G$ is finitely presented.

Let $G$ be a group and $N\triangleleft G$. Show that if $G/N$ and $N$ are finitely presented, then $G$ is finitely presented. Worked on this problem for about 2 hours before we all threw in the towel....
5
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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 $...
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0answers
52 views

Classifying automorphisms using a group presentation.

I'm new to group presentations and after some playing around with the concept, I've tried to find some relatively clear criteria in terms of the defining relations, that would tell us if a map is an ...
5
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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 ...
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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
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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
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0answers
78 views

Residually finiteness for a factor group

Suppose we have a finitely presented residually finite group $G=\langle X\,; R \rangle$, two isomorphic finite subgroups $C$ and $D$ of $G$. The question is whether the group $H=\langle X\,; R, C=D \...
5
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0answers
292 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 ...
4
votes
4answers
614 views

Presentation of a non-trivial group

I'm having a bit of trouble understanding group presentations. For example, I'm reliably informed that the group $$ \langle x, y \mid x^2=y^3 \rangle $$ is not the trivial group, but I don't see ...
4
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2answers
933 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.
4
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3answers
115 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 ...
4
votes
3answers
179 views

Prove G is a nonabelian group of order 20

Given that $G = \langle x,y | x^5=y^4=1,yx=x^2y\rangle$, how would I prove $G$ is a non-abelian group of order $20$ (and not isomorphic to $D_{10}$)? Here's what I have so far: $y^4=1$ so $xy = y^...
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=...
4
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3answers
496 views

A group presentation for $\mathbb{Z}_2\times \mathbb{Z}_2$

I know that the only groups of order 4 are $\mathbb{Z}_2\times \mathbb{Z}_2$ and $\mathbb{Z}_4$ up to isomorphisms. And I also know that the group presentation of $\mathbb{Z}_4$ is $\left ( a:a^4=1 \...
4
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2answers
118 views

Two questions about groups and presentations

i have two questions about group presentations: Given the presentation $\langle a,b:a^2=1=b^3,(ab)^3=1\rangle$. I know that this is the presentation of $A_4$ but how to deduce that. Should i give a ...
4
votes
3answers
211 views

Getting the wrong order of a finitely presented group

Let $G=\langle x, y \mid x^4=y^3=1, y^{-1}xy=x^{-1}\rangle$. What is $G$? I started by taking $$y^2=y^{-1}xy y^{-1}x^{-1}= (y^{-1}xy) y^{-1}x^{-1}=x^{-1}y^{-1}x^{-1}=x^{-1}y^{-1}(y^{-1}xy)=x^{-1}...
4
votes
1answer
115 views

Proving that an element of a given group has an infinite order

I am given the following group: $$G = \langle x_1,x_2,x_3 | x_1^2 = x_2^2 = x_3^2 = e, \langle x_1, x_2 \rangle = \langle x_2, x_3 \rangle = e \rangle,$$ where $\langle a,b \rangle = e$ is the triple ...
4
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2answers
95 views

Which group is this?

Define $G=\left\langle a,b\ |\ a^2=1, (ab^2)^2=1 \right\rangle $. This is an infinite group whose Cayley graph is best described as a two-dimensional grid. Is it a well-known group? What is known ...
4
votes
4answers
359 views

Upper bound for order of finite group given relations

Say I have a group with the following presentation: $$ G = \langle a,b \mid a^2 = b^3 = (ab)^3 = e \rangle $$ During a conversation someone had mentioned that the order for $G$ must be less than or ...
4
votes
1answer
310 views

Presentation of a group

Let $a, b \in \mathbb{N},\ a, b\neq 0$ such that $(a,b)=1.$ Suppose $G$ is a group with presentation $$ G=\langle x, y \mid x^{-1}y^{-1}xy^{a+1}=1,\ y^{-1}x^{-1}yx^{b+1}=1\rangle. $$ Prove that $G= \...
4
votes
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?
4
votes
2answers
555 views

Examples of non-finitely presented groups

I know several constructions leading to finitely generated non-finitely presented groups, using amalgamated products: Property: Let $A,B$ be two finitely presented groups. Then $A \underset{C}{\ast}...
4
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2answers
93 views

Unprovability of $i^2 =1$ from $\langle i \mid i^4 =1\rangle$ and similar problems

This question is related to Can I derive $i^2 \neq 1$ from a presentation $\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle$ of Quaternion group $Q$? I know I'm going too far but let me just ask... ...
4
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1answer
246 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
4
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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 ...
4
votes
1answer
76 views

Is there a way to describe the structure of $Aut(UT(3, p))$?

Is there a way to describe the structure of the automorphism group of $$C_{p}^2 \rtimes C_p \cong \langle x, y, z | [x,y]=z, [x,z]=[y,z]=x^p=y^p=z^p=e \rangle \cong UT(3, p)?$$ Here $p$ is an odd ...
4
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1answer
66 views

Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$

When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following: $\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$ See https://groupprops....
4
votes
1answer
201 views

Presentation of wreath product $G=S_3 \wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?

I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows; Let $G = S_3 \wr S_3$, the permutational wreath product of two symmetric groups of ...
4
votes
1answer
150 views

Group Presentation - Categorical Interpretation

In this post, user Martin Brandenburg wrote $$\langle x_1,x_2,\dotsc \mid R_1,R_2,\dotsc \rangle$$ is defined to be a group satisfying the following universal property: The group has elements $x_1,...
4
votes
1answer
312 views

Understanding group structure using GAP

My question is regarding how to understand structure of group using GAP. Using StructureDescription(G), GAP gives some information like it is direct product or ...
4
votes
1answer
109 views

Presentation of groups and positive expressions

For a group $G=〈S|R〉$, $S,R$ are both finite. A positive element $g\in G$ is an element of $G$ that can be written as a finite product of elements of $S$ only. A positive expression of $g$ is a word $...
4
votes
1answer
86 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
4
votes
1answer
147 views

Groups with a R.E. set of defining relations

Reading around I found the following two assertion: 1) Every countable abelian group has a recursively enumerable set of defining relations. 2) Every countable locally finite group has a recursively ...
4
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1answer
88 views

Is there an algorithm to solve all soluble group word problems?

What I mean is, is there an algorithm that given any finitely presented group with soluble word problem can solve the word problem on that group?
4
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1answer
163 views

Group Presentations and Cayley graphs

I am trying to understand group presentations and Cayley graphs, and have a few questions I am confused about. Let $G=(V,E)$ be a finite $d$-regular graph that is known to be a Cayley graph for the ...
4
votes
1answer
523 views

how to prove two groups are NOT isomorphic?

I have two groups defined by presentations $$\langle x, y \mid x^p = y^q \rangle$$ $$\langle x, y \mid x^{p'} = y^{q'} \rangle$$ where $p,q,p',q'$ are all integers greater then $1$, and $\gcd(p,...
4
votes
1answer
86 views

What is this presentation isomorphic to?

I am trying to determine the group given by the presentation: $\langle\ a, b, c\ \vert\ a^2=b^5,\ b^2=c^3,\ c^2=a^7\ \rangle$. I've been trying to tackle this problem by starting with the first ...
4
votes
2answers
104 views

Relations in Group Presentation

In an introduction to abstract algebra, I was recently introduced to the idea of presenting a group - minimally, a group is just a set of generators along with a set of relations amongst the ...
4
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1answer
187 views

Describing the group defined by $\langle a,b \mid (ab)^2=(abaa)^2=(abbb)^2=e\rangle.$

I'm trying to find an isomorphism of a group with the following presentation:$$\langle a,b \mid (ab)^2=(abaa)^2=(abbb)^2=e\rangle$$ Basically, I'm not that experienced with groups so I'm wondering if ...
4
votes
1answer
134 views

Identifying the group $G = \langle x,y \space | \space xy=yx, x^4=y^2 \rangle$ from the given presentation [duplicate]

I'm trying to solve a problem in my textbook which asks me to identify the groups $G_1 = \langle x,y \space | \space x^3y=y^2x^2=x^2y\rangle$ and $G_2 = \langle x,y \space | \space xy=yx, x^4=y^2 \...
4
votes
1answer
86 views

Showing $\langle a,b\mid abab^{-1}\rangle$ and $ \langle c,d \mid c^2d^2\rangle$ are isomorphic.

I computed the fundamental group of the Klein bottle in two different ways and obtained two seemingly different answers: $$ \langle a,b \mid abab^{-1}\rangle $$ and $$ \langle c,d \mid c^2d^2\rangle. $...
4
votes
1answer
114 views

How to convert one presentation into another? Please explain using a dihedral group as an example.

How can we convert a given presentation of a group $G$ into an another presentation? Would anyone please explain to me by converting two different presentations of a dihedral group? Thanks in ...