# Questions tagged [group-presentation]

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

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### 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 $m,n,p>1$ , ...
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### 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 ...
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### 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) ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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... ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...