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

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3
votes
2answers
59 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 ...
2
votes
1answer
71 views

Trying to understand proof of isomorphsim between two group presentations

I am trying to understand a proof given of an isomorphism between an infinite and finite presentation of Thompson's group F in the following paper by Cannon, Floyd and Parry. ...
-2
votes
1answer
104 views

Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
2
votes
1answer
41 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 ...
4
votes
4answers
381 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 ...
5
votes
1answer
92 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?
1
vote
2answers
91 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 Let's take ...
0
votes
1answer
28 views

Is $G=\langle g,h:g^{2p}=h^2=1,g^h=g^{-1}\rangle$ for $|G|=\text{ord }g\cdot\text{ord }h$ a valid group presentation?

Let $G$ be a finite group of order $4p$ where $p\ge 5$ is any odd prime number and $\sigma,\tau\in G$ with $\text{ord }\sigma =2p$, $\text{ord }\tau=2$ and $\sigma^\tau :=\tau\sigma \tau ...
1
vote
1answer
45 views

Show that $\langle a,b,c;a^3,b^2,ab=ba^2 , c^2, ac=ca,bc=cb \rangle$has order 12

Show that $\langle a,b,c;a^3,b^2,ab=ba^2 , c^2, ac=ca,bc=cb \rangle$has order 12 and find the permutation group isomorphic to it! I know that $S_3$ is presented by $\langle ...
2
votes
2answers
134 views

Show that $S_3$ is presented by $\langle a,b\mid a^3, b^2,ab=ba^2\rangle$

Show that group $S_3$ of the objects $x,y,z$ is presented by $\langle a,b \mid a^3,b^2,ab=ba^2\rangle$ under the mapping $a \to (xyz)$ , $ b\to (xz)$ I'm confused to what is to be shown in these ...
2
votes
2answers
88 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
125 views

Group presentations

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
59 views

$\mathbb{C}[x,y\,|\,x^m=1,y^n=1]\cong\mathbb{C}[z\,|\,z^{mn}=1]$ as complex algebras?

If $G$ is any finite abelian group and $K$ an algebraically closed field with $|G|\neq 0$ in $K$, then the group algebra $K[G]\cong M_{n_1}(K)\times\cdots\times M_{n_k}(K)$ by Maschke's and ...
0
votes
3answers
223 views

Find a set of generators and relations for $S_3$

Can you help me with this? By trial, I came up with the generators and relation. However, how do I prove that the generators and relations uniquely determine $S_3$? Problem Find a set of generators ...
2
votes
3answers
91 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 ...
2
votes
1answer
78 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} } ...
5
votes
3answers
132 views

What is this group? (Recognising a group from a presentation).

I am trying to find out what the following group is: $$G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3\rangle.$$ Due to the isomorphism problem for groups, there is no algorithmic way to approach ...
2
votes
1answer
48 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 ...
1
vote
0answers
62 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 ...
2
votes
1answer
69 views

Dehn and Wirtinger Presentations of Knot Groups and their connection

I'm currently working through N.D. Gilbert and T. Porter's Knots and Surfaces. In it the idea of a Wirtinger presentation and a Dehn presentation for a group associated with a given knot is ...
3
votes
2answers
70 views

Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
15
votes
1answer
280 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
1
vote
0answers
68 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 ...
0
votes
1answer
39 views

The deficiency of surface group

Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The ...
2
votes
0answers
36 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 ...
4
votes
1answer
80 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 ...
5
votes
2answers
110 views

Finitely generated group which is not finitely presented

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 ...
3
votes
0answers
140 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
4
votes
1answer
72 views

Showing a group is isomorphic to a group with known presentation

Let $H$ be a group with presentation $\langle h_1, \dots, h_n \mid r_1 = \dots = r_m = 1\rangle$. If there are $g_1, \dots, g_n \in G$ which satisfy the relations $r_1, \dots, r_m$, when is $\varphi : ...
5
votes
1answer
171 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 ...
3
votes
3answers
107 views

What should be the presentation of $\mathbb Z$?

In the Dummit-Foote text the definition of relation and presentation (Group theory) are introduced as: In connection with the above definition I wounder what should be the presentation of ...
1
vote
1answer
93 views

Group presentations and subgroups

How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$? $\bf Edit$: Given that $|G|=12$.
5
votes
5answers
193 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 ...
6
votes
0answers
93 views

Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with 2 generators. More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of ...
5
votes
2answers
132 views

notations of generators and relations

I need to understand the following about generators and relations notations: Is $\langle a,b \mid a^kb^l\rangle =\langle a,b\mid a^k=b^l\rangle =\langle a,b\mid a^k,b^l\rangle$? Is $ \langle a,b\mid ...
1
vote
1answer
115 views

presentation of a group

Let $G=F( x,y)$ be a free group with two generators, assume that $H\leq G$ where $H=\langle xyxy^{-1}\rangle $, let $N$ be the normal closure of $H$. Is it true that $G/N$ has a presentation $\langle ...
3
votes
1answer
131 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
35 views

presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
3
votes
2answers
74 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... ...
2
votes
0answers
82 views

Representing digraphs by undirected graphs

One can represent every group as a directed graph with colored edges (its Cayley graph). Identifying the colors of the edges with specific vertices of the graph (its generators), one ends up with a ...
1
vote
2answers
117 views

Magma say the size of my group is $0$

I am trying to define a group in terms of generators and relations in Magma and check its size, but Magma says the size of my group is $0$. The same code works for smaller presentaions of groups. What ...
5
votes
2answers
117 views

Is there software to help with group presentation

I wrote a computer program that generates group presentations. I would like to know the sizes of the resulting groups. I know that this is undecidable. Are there good heuristic programs that can ...
1
vote
2answers
43 views

A translation and a negation in $\mathbb{C}$ generate the infinite dihedral group.

I'd like to show that the linear functions $$ \varphi(z) = z+b, \;\;\; 0\neq b\in \mathbb{C}$$ $$ \psi(z) = -z+c, \;\;\; c\in \mathbb{C}$$ generate, under composition, a group isomorphic to ...
4
votes
0answers
53 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 ...
1
vote
1answer
48 views

How can i create a presentation of a group ?

in Dummit and Foote , the notion of presentation is introduced in section 1.2 which talks about dihedrial group of order $2n$. and after this , it was rare to talks about presentation throw the ...
1
vote
1answer
48 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\\ ...
1
vote
2answers
102 views

tensor product and direct product of algebra presentations

Let $R$ be a commutative unital ring and $R\langle x_i\mid f_j\rangle$ denote a unital $R$-algebra presentation. Q1: What is the presentation of $R\langle x_i\mid f_k\rangle\otimes R\langle y_j\mid ...
5
votes
1answer
235 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 ...
1
vote
2answers
142 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
5
votes
1answer
93 views

How to prove these two groups are isomorphic

If $G_{1}$ is $\langle a,b \mid a^2 = b^2 \rangle$ and $G_{2}$ is $\langle p,q \mid pqp^{-1} = q^{-1} \rangle$, find an isomorphism $\phi : G_{1} \rightarrow G_{2}$. I tried the obvious by letting ...