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Questions tagged [group-presentation]

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

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How do I prove that group determined by generators and relations is trivial? [closed]

I have a group $\langle a,b\mid a^4=b^2=1, ab^2=b^3a, ba^3=a^2b\rangle $ how do I show that it's trivial?
2
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0answers
67 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
134 views

Is there an algorithm/method for actually drawing Van Kampen diagrams?

In theory, given a presentation for a group, I understand what a Van Kampen diagram should look like, but I struggle when it comes to the practicalities of drawing it. Is there an algorithm/method ...
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.
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1answer
50 views

Notational confusion about HNN-extensions: $G=K \ast_{H,t}$.

Let $G=K \ast_{H,t}$ denote an HNN-extension, i.e., $$H \le K \le G, H^t \le K.$$ Is it true that $\{K, t\}$ is a generating system for $G$? In particular, $G/K$ is cyclic?
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 ...
1
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1answer
52 views

Presentations of subgroups

I've been looking into finding the presentation of subgroups $H$ of $G$ from a known presentation of the group $G$. I know that usually, removing generators but keeping the set of relations the same ...
3
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1answer
89 views

What is the name of this normal subgroup of the free group?

Given the free group $F_S$, we can define a normal subgroup of $F_S$ as follows. For a given element $w \in F_S$ and $s \in S$, define "the count of $s$", $c_s(w)$, as the number of times $s$ appears ...
5
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0answers
51 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 ...
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1answer
64 views

How to show that elements in these groups pairwise commute? [closed]

Let $F_2$ be the free group on $x,y$ and $N$ the normal subgroup generated by $x^2,y^4,xyx^{-1}y^{-1}$. Consider the group $F_2/N$ and two elements $xN$ and $yN$ in it. How do I show that elements in ...
2
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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 $m=13$....
3
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1answer
171 views

Does $SL_2(\mathbb{Z}[\sqrt{2}])$ have a finite presentation?

The modular group group $\text{PSL}_2(\mathbb{Z})$ can be written as something that is nearly a free group on two elements: $$ SL_2(\mathbb{Z}) \simeq \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/3\mathbb{Z}...
3
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2answers
97 views

Find the group given the presentation

I have the group presentation $$ G := \langle a,b |a^8=b^8=1,a^{-1}ba=b^{-1},b^{-1}ab=a^{-1} \rangle. $$ Which group is it? Notably, what about its order?
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0answers
63 views

Centralizers of non-central elements of a special group

Let $G$ be a finite group so that $\frac{G}{Z(G)}\cong \mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime. Then $\frac{G}{Z(G)}$ has the presentation $\langle aZ(G),bZ(G)|a^p ,b^p , aba^{-1}b^{-1}...
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0answers
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Isomorphy of simple groups of order 360 : a proof with a presentation

It is well known that all simple groups of order 360 are isomorphic with the alternating group $A_{6}$. Cole's original proof is here on StackExchange : $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ ...
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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 ...
4
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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^...
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2answers
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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} =...
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0answers
79 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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1answer
31 views

Terminology: reversing the order of the generators.

Let $H=\langle Y\mid S\rangle$ be a finite group. Then, for all $h\in H$, we can write $$h=y_1^{\eta_1}\dots y_m^{\eta_m}$$ for some finite $m\in\Bbb N$ and finite $\eta_i\in\Bbb Z$, for $y_i\in Y$....
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2answers
90 views

Automata defined by group presentations.

I apologise if this question is ill-formed or too broad. It's just for fun. I've thrown in the soft question tag for good measure. Let $G=\langle X\mid R\rangle$ be a group. What can be said in ...
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0answers
63 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
votes
1answer
91 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
votes
2answers
75 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 ...
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1answer
71 views

Order of group $\langle x,y,z \mid x^2=y^2= z^2= e, xyz=yzx = zxy \rangle$

Group $\langle x,y,z \mid x^2=y^2= z^2= e, xyz=yzx = zxy \rangle$ has order 16. I put $a = xy$ and we have $za=az=yzay$ but I can't continue it.
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0answers
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Show that a group is trivial [duplicate]

Let $G$ be the group generated by $a,b,c$ with relations $ab=b^2a$, $bc=c^2b$, and $ca=a^2c$. Show that $G$ is trivial. A related problem is that for $G_1$ generated by $a,b,c,d$ with similar ...
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0answers
123 views

Understanding the Todd-Coxeter Coset Enumeration Algorithm.

I've just finished reading Chapter 6 of "Presentations of Groups," by D. L. Johnson. The Details: Quoting Johnson . . . Definition 1: Let $\langle X\mid R\rangle$ be a finite group and put $F=\...
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votes
2answers
90 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 ...
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3answers
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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 ...
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1answer
103 views

Finding the invariant factors of $G/G'$.

I'm reading "Presentations of Groups," by D. L. Johnson. Tl;dr: How do you find the invariant factors of $G/G'$ from a presentation of a group $G$? The Details: In Chapter 3 of the book ibid., ...
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0answers
53 views

Presentations of groups

I'm trying to understand presentations of groups better than I currently do. Does anyone know of any good online sources that are worth a look?
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1answer
26 views

Conditions, under which a mapping of generators of a group induces an automorphism.

Throughout this question I will presume the following: $G$ is a finite group generated by two elements $a,b$, and $f: \{a,b\} \rightarrow G$ is a function, such that $f(a),f(b)$ generate $G$. - ...
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 $g$ ...
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1answer
117 views

Identifying $\langle \{x_h\mid h\in H\}\mid \{x_{h'}x_{h'a}x_{h'b}^{-1}\mid h'\in H\}\rangle$ if $H=\langle a,b\mid a^2, b^4, ab=ba\rangle$.

Let $H$ be $C_2\times C_4=\langle a,b\mid a^2, b^4, ab=ba\rangle$ with identity $e$. What group is $$G_H(a, b):=\langle \{x_h\mid h\in H\}\mid \{x_{h'}x_{h'a}x_{h'b}^{-1}\mid h'\in H\}\rangle?$$ ...
1
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1answer
70 views

Identifying $\langle r,s\mid srs^{-2}r, r^{-1}srsr^{-1}\rangle$.

What group is $$G:=\langle r,s\mid srs^{-2}r, r^{-1}srsr^{-1}\rangle?$$ Thoughts . . . Using IdGroup in GAP on G with ...
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2answers
122 views

Given a presentation of a group, how do you find what it is isomorphic to?

Suppose you have a group presentation $G=<a,b|a^{5}=b^{2}=e, ba=a^{2}b>$. In general, how do you find the group it is isomorphic to? I've seen examples using the Fundamental Theorem of Finitely ...
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1answer
68 views

Index of a subgroup with GAP

Let $G=\langle x,y\ |\ x^2=y^4,\ xyxy^6=1 \rangle$ be a finite presentation group and $H=\langle x^4,y^4,xyx^{-1}y^{-1}\rangle$ be a subgroup of $G$. How could we obtain the index of $H$ in $G$ with ...
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1answer
71 views

Is there a quicker way to do this in GAP?

If this isn't clear, I'm sorry. I'm investigating what groups one gets using a certain procedure. Take a group $H$. Let $w=x_ex_mx_k^{-1}$ be an element of the free group on $X:=\{x_h\mid h\in H\}$ ...
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2answers
52 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 ...
0
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1answer
56 views

Identifying $\langle x_1, \dots, x_6\mid x_1x_2=x_3, x_2x_3=x_1, x_3x_1=x_2, x_4x_5=x_6, x_5x_6=x_4, x_6x_4=x_5\rangle.$

Consider the presentation $$G:=\langle x_1, \dots, x_6\mid x_1x_2=x_3, x_2x_3=x_1, x_3x_1=x_2, x_4x_5=x_6, x_5x_6=x_4, x_6x_4=x_5\rangle.$$ What group is it? Thoughts: Calculations in GAP show that ...
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1answer
50 views

Identifying $\langle x_1, x_2, x_3, x_4\mid x_1x_2=x_3, x_3x_2=x_1, x_1x_4=x_3, x_3x_4=x_1\rangle$.

What group is $\langle x_1, x_2, x_3, x_4\mid x_1x_2=x_3, x_3x_2=x_1, x_1x_4=x_3, x_3x_4=x_1\rangle$? Thoughts: The group is infinite according to GAP, $x_2=x_4$, and $x_2^2=id.$
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1answer
39 views

Identifying $\langle x_1, x_2, x_3, x_4\mid x_1x_2=x_3, x_2x_1=x_3, x_1x_2=x_4, x_2x_1=x_4\rangle$.

What is the group $\langle x_1, x_2, x_3, x_4\mid x_1x_2=x_3, x_2x_1=x_3, x_1x_2=x_4, x_2x_1=x_4\rangle $? It has infinite order according to GAP and the generators $x_3, x_4$ are redundant. Is it ...
1
vote
0answers
71 views

Proving a cyclically presented group is infinite using GAP.

I apologise if this is too broad or is otherwise off topic. The Problem: I'm looking for ways to prove that a cyclically presented group, given in terms of generators and relators, is infinite using ...
1
vote
1answer
61 views

Showing that a subgroup of $GL(2,\mathbb{C})$ is defined by a certain presentation

I need to show that the subgroup of $GL(2,\mathbb{C})$ generated by $$a:=\begin{pmatrix}w^2 & 0 \\ 0 & w\end{pmatrix}$$ where $w=e^{\frac{2\pi i}{3}}$ and $$b:=\begin{pmatrix}0 & i \\i &...
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1answer
69 views

Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$ [closed]

Could someone give me a suggestion to solve this problem? Show that $\langle{x,y\,\vert\,x^{2}y^{-2}}\rangle$ is not isomorphic to $S_{3}$
1
vote
4answers
84 views

Prove that $\langle{a,b\,\vert\,aba^{-1}b^{2}}\rangle$ is not isomorphic to $\mathbb{Z}/3\mathbb{Z}$.

Could anyone give me a suggestion to solve this problem about group presentations? Prove that $\langle{a,b\,\vert\,aba^{-1}b^{2}}\rangle$ is not isomorphic to $\mathbb{Z}/3\mathbb{Z}$
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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.
0
votes
1answer
287 views

Presentation of the group of Quaternions

Reading Rotman's book on group theory he states that the following is a presentation of the group of quaternions: $$Q' = \langle x,y \mid x^{2}=y^{2}, \, xyx=y \rangle$$ My question is: how does one ...
1
vote
1answer
70 views

Computing the order of a particular group from its presentation

Suppose that $m \geq 3$, $\mathcal{F}$ is the free group whose basis is $\{a,b\}$, and H is the normal subgroup of $\mathcal{F}$ generated by $\{bab^{-1}a, a^{2^{m-1}}, b^{-2}a^{2^{m-2}}\}$. Let $G_m =...
1
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1answer
74 views

Finding groups from presentations

I am trying to solve the following problem: Find all groups with two generators $a$ and $b$ in which $a^4 = 1, b^2 = a^2, $ and $bab^{-1} = a^{-1}.$ I know that this is a presentation of the ...