Questions tagged [group-presentation]

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

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85 views

Showing affine transformations group generated by $2x$ and $x+1$ is the Baumslag-Solitar group.

I want to compute the presentation groups of $\langle f,g\rangle$ the generated group of affine transformations with $f(x)=2x$ and $g(x)=x+1.$ The affirmation is $\langle f,g\rangle=\langle a,b\...
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1answer
72 views

Uniqueness of group given only group presentation

I'm trying to understand how a given presentation of a group is well defined. It says on Wikipedia (https://en.wikipedia.org/wiki/Presentation_of_a_group) that $G \ =\ \langle a\ |\ a^n=1\rangle$ is a ...
2
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2answers
77 views

Proof of a relation of Braid groups

Let $B_n$ be the braid group on $n$-strings, generated by $\alpha_1,\ldots, \alpha_{n-1}$ with relations $\alpha_i \alpha_j = \alpha_j \alpha_i$ for $|i-j|>1$ and $\alpha_i \alpha_{i+1} \alpha_{i} =...
2
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1answer
21 views

$G=\langle S|R\rangle$ is finitely presented simple, then if $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$

I'm having some trouble seeing $G=\langle S|R\rangle$ is finitely presented and simple, then if the world in free group $F(S)$, $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$ ...
2
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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
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2answers
78 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...
2
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1answer
696 views

Presentation of Heisenberg Group $\mathbb{H}$ over the field $\mathbb{F}$

Let $\mathbb{F}$ denote the finite field. Denote $\mathbb{H}_{\mathbb{F}}=\left\{ \left( {\begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} } \...
2
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1answer
38 views

Prove that if $H,K \leq G$ where $G$ hyperbolic, $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate

Let $G = \langle S \mid R \rangle$ be a finite presentation, $G$ is $\delta$-hyperbolic. Prove that if $H,K \leq G$ where $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate. I am ...
2
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1answer
84 views

How to show that $|D_{2n}| = 2n$ via the presentation?

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
2
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1answer
45 views

How to reconcile two different computed first homology groups for this space?

I am asked to classify the compact surface obtained by pasting the edges of a polygonal region with the labeling scheme $abcdabdc$ and compute it's first homology group. I classify the space as the ...
2
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1answer
77 views

Identifying group structure of matrix group

I am working on classifying the stabilisers of quadratic binomials in $GL(\mathbb{C}^{n})$ but struggling to identify the groups which are appearing. One example is the binomial $$x_{1}x_{2}-x_{1}x_{...
2
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1answer
68 views

If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is $0$.

I need a reference for the following theorem: If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is also $0$. I think it's by ...
2
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1answer
96 views

Finding presentation of a subgroup in GAP

I have a finitely prsented group $G$ and its subgroup $H$. They aren't stored however as fp groups in GAP. I can quite easy obtain some presentation $pr$ of $G$ in. How can I obtain the presentation ...
2
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1answer
113 views

Higman's group is not trivial.

I'm trying to prove that Higman's group is not trivial. In order to do that, first of all I have to define the following groups: $ \langle h_{i},h_{i+1}| h_{i+1}h_{i}h_{i+1}^{-1}=h_{i}^2\rangle$ for ...
2
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1answer
31 views

$\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
2
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1answer
316 views

Surjective Group Homomorphism From Braid Group Into Symmetric Group

I am reading this article on wikipedia. Here's the relevant excerpt: By forgetting how the strands twist and cross, every braid on $n$ strands determines a permutation on $n$ elements. This ...
2
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1answer
294 views

Tietze transformation

I have a question in which I have to transform $\textbf{I.}$ $\langle a,b,c \mid b^2, (bc)^2\rangle$ to $\textbf{II.}$ $\langle x,y,z\mid y^2, z^2\rangle$ using Tietze transformations. My ...
2
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1answer
198 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
2
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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 ...
2
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1answer
52 views

Elements of a group

Consider the following presentation for $A_4$ $$< p, q\, |\, p^2 = (pq)^3 = q^3 =e>$$ There exist eight elements of order $3$ in this group. Deduce these elements by writing them as products of $...
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2answers
653 views

Group presentations - again

My question is about finding presentations for finite groups. It's along similar lines to my earlier question -- but is subtly different! The earlier question is here Group presentations: What's ...
2
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1answer
99 views

Prove the equivalence of presentation of a free group with a free product

I want to prove that the following presentation of a free group (generators and relations): $$\left(\begin{array}{c|c} x_0,a_0&a_0a_1x_2=x_0a_1\\ x_1,a_1&a_1a_2x_0=x_1a_0\\ x_2,a_2&...
2
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1answer
70 views

Proving that an element in an algebra presentation is nonzero

Let $F$ be a field, $F\langle x,y\rangle$ the free $F$-algebra on two generators (polynomials in two noncommuting variables), $A= F\langle x,y\,|\, xy\!=\!1\rangle= F\langle x,y\rangle/\langle\langle ...
2
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1answer
79 views

concrete equality of group presentations

Why does this equality hold? $$\begin{array}{rl} \langle a,c| &ca^{-1}cac^{-1}aca^{-1}c^{-1}ac^{-1}a^{-1}ca^{-1}c^{-1}a,\\ &ac^{-1}aca^{-1}cac^{-1}a^{-1}ca^{-1}c^{-1}ac^{-1}a^{...
2
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1answer
92 views

Presentations of subgroups of S6 as permutations

Computer science professor, self-taught abstract algebraist. Beginner with GAP and SAGE. Can someone show me the quickest way to, when given the description of a subgroup of S6, obtain its ...
2
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2answers
50 views

Group size as a function of presentation length?

Given a presentation of a group, together with the promise that the group is finite, is there a computable upper bound on the size of the group? Edited to add: by "presentation" we mean a set of ...
2
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1answer
70 views

Understanding groups that have a similar presentation as $S_n$

It is well known that the symmetric group $S_n$ can re presented using the generatos $\sigma_1, \dots, \sigma_{n-1}$ subject to the relations: $$\sigma_i^2=1,\\ \left[\sigma_i ,\sigma_j\right] = e \...
2
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2answers
77 views

Presentation of $D_4$

When one writes $D_4=\langle r,s\mid r^4=s^2=1,rsrs=1\rangle$ they are describing a quotient group. Let $S=\{s,r\}$ and $R=\{r^4,s^2,rsrs\}$. $$F_S=\langle r,s\rangle,\quad R^{F_S}=\{grg^{-1}\mid r\in ...
2
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1answer
49 views

Presentation of $D_4 \rtimes Aut(D_4)$

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the presentation of $D_4 \rtimes Aut(D_4)$ is as follows. $$ D_4 \rtimes Aut(...
2
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1answer
85 views

Finitely presented monoid with non-solvable word problem

A. Markoff (On the impossibility of certain algorithms in the theory of associative systems) and E. Post (Recursive Unsolvability of a Problem of Thue) provide examples of a finitely presented monoid ...
2
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1answer
37 views

For a Group $G$ with$S\subseteq G$, when does $ \left\langle S \right\rangle $ equal the normal closure of $S$?

To obtain the group with the presentation that has all elements of a subset $S$ equal to the identity, you must take the quotient of the free group (on the appropriate number of generators) that sets ...
2
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1answer
73 views

What are the conjugacy classes of the group $\{k,r \,| \, k^3=r^2=(kr)^4=1\}$?

How can I determine the conjugacy classes of a group if I have the presentation of the group? For example, we know that $S_4$ has the presentation $$ S_4=\{k,r \,| \, k^3=r^2=(kr)^4=1\}. $$ What are ...
2
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1answer
83 views

Modify a Dehn presentation

Suppose you have a Dehn presentation $\langle X \mid R \rangle$ of (say not the free group) a hyperbolic group. Has there been some work done on changing this presentation, e.g. adding a relation ("...
2
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1answer
69 views

Is this group simple?

I have this group presentation $G = \langle a,b | ba = a^{-1}b\rangle$. I'm wondering if this group is solvable or not. I tried to show that this group is simple, because if it is, then it can not be ...
2
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1answer
544 views

Presentation of Abelianization of a group

Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...
2
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1answer
86 views

Why $G$ is not free?

I have to show that $G=\langle x,y,z\ |xz=zx \rangle$ is not free. Now either I show it has an element of finite order which I dont see works here or I show it has no non-trivial defining relators i.e....
2
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1answer
76 views

The order of the group $\langle a, b| w^3, w\in\langle a, b\rangle\rangle$ for $w$ being any word.

I came across this group mentioned in passing as finite. Does anyone know the order of the group, and where I can find a proof of this quantity? Replacing $3$ with $n$, does this problem have a name ...
2
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1answer
202 views

Free presentations of $\mathbb{Z}G$-modules

Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free ...
2
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0answers
57 views

What does Sylow theory have to say about group presentations?

What does Sylow theory have to say about group presentations? Of the books on combinatorial-group-theory I have looked in so far, the following do not contain any reference to Sylow's Theorems: ...
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0answers
27 views

Symmetrising the relations in a presentation of a group

Let $G$ be a finitely presented groups defined by $$G=\{x_1,\ldots,x_n\mid R_1(x_1,\ldots,x_n)=\cdots=R_m(x_1,\ldots,x_n)=1 \}.$$ Let this presentation be denoted by $P$. Let $S_n$ be the ...
2
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0answers
303 views

Cayley graph of Rubik's cube group

(a) I would like to know whether there is a group theoretic approach for calculating the diameter of the Cayley graph of Rubik's Cube group. I know it's been proved that the above diameter is $20$ ...
2
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0answers
69 views

Two presentations of a group, one certainly finite. Need the other be?

I know the answer to the question above is "no", quite flatly. The counter example is below: $$\mathbb{Z}\cong\langle a,b\mid b^2a^{-1}\rangle\cong \langle a,b\mid\lbrace b^{2^{n+1 }}a^{-2^n}:n\in\...
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0answers
61 views

How to prove that two groups with different presentations are isomorphic in a naive way?

One can define a presentation of a group naively (ala Dummit-Foote in Chapter 1.2), i.e., as a group generated by certain elements with certain relations such that all other relations follow from the ...
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0answers
112 views

Reference request fundamental group of surface of genus $g$ and $n$ boundary components

Let $\Sigma_{g,n}$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $\partial_i$. I would like to find a reference for the ...
2
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0answers
54 views

Burnside groups : a few questions

Let $G=B(n,e)=F_n/\langle\langle F_n^e\rangle\rangle$ be the Burnside group on $n$ generators with exponent $e$, i.e. the quotient of the free group on $n$ generators $F_n$ by the normal subgroup ...
2
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0answers
70 views

Free group and isomorphism

suppose $F$ is a free group generated by $x$ and $y$. Prove that $u=x^2, v=y^3$ generates a subgroup of $F$, and it is isomorphic to the free group on $u,v$ Prove that $u=x^2, v=y^2, z = xy$ ...
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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 $m=13$....
2
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0answers
80 views

Is there a general way to simplify such group presentations (Free Abelian Group with Relations)?

If I want to simplify the group presentation (free abelian group with relations) $$\langle a,b,c\mid 2a=b=2c\rangle,$$ I can simplify it as $$\langle a,c\mid 2a=2c\rangle\cong\langle a,a-c\mid 2(a-c)...
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0answers
64 views

What is the intersection of the vertices of a face of a simplicial complex?

I am currently reading "Subgroup graph methods for presentations of finitely generated groups and the contractibility of associated simplicial complexes" By Cora Welsch and I'm a bit stuck with ...
2
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0answers
62 views

Is every finite group an arithmetic group?

Lubotzky-Phillips-Sarnak(LPS88) constructed Ramanujan graphs of degree $d=p+1$ (for an odd prime $p$) as Cayley graphs of projective linear groups with respect to a carefully chosen set of generators ...