Questions tagged [group-presentation]

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

6
votes
1answer
617 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 $G$. ...
3
votes
3answers
638 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 $\mathbb Z$...
1
vote
1answer
170 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
6answers
367 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 d^2 =...
12
votes
0answers
186 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 ...
6
votes
2answers
598 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
votes
1answer
227 views

Is it true that $G/N$ has a presentation $\langle x,y\mid xyxy^{-1} \rangle $? [closed]

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 ...
6
votes
0answers
156 views

Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
3
votes
1answer
781 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 ...
3
votes
0answers
134 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 ...
4
votes
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... ...
2
votes
0answers
245 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
257 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
564 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 ...
4
votes
2answers
930 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.
1
vote
2answers
54 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 $Dih_\...
5
votes
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 \...
1
vote
1answer
77 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 ...
2
votes
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&...
1
vote
2answers
223 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 ...
9
votes
1answer
2k 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
169 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 ...
1
vote
1answer
99 views

The automorphism of a group when given a group presentation

How do you find the automorphism group when given a group presentation? See the comments for an attempt to answer by the user asking this question.
3
votes
5answers
1k views

Definition of presentation of a group.

basing myself on suggestions I found in previous discussions, I have opted for the algebra book of Dummit. However, I have now a little problem. On page 26 (3th edition) the authors say that "...
5
votes
1answer
110 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 $a$...
1
vote
2answers
81 views

Choosing central generator for nilpotent group generated by 3 elements

Let $G$ be a group, which is 2-step nilpotent torsion-free generated by three elements (in a minimal presentation) such that the centre of $G$ is generated by one element (i.e. $C(G)$ is infinite ...
0
votes
2answers
90 views

Group presentation

Given $G = \langle x,y \mid x^{16}=y^{24}=e, x^2=y^3\rangle$, if $G$ is abelian, find its order. Note: An estimation gives $\left|G\right| \leq 48$, but is there a quotient of $G$ with order $48$? (...
2
votes
2answers
539 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
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 ...
3
votes
1answer
1k views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by $$D_n=\{I,...
8
votes
0answers
581 views

Way to Tietze's Transformation Theorem

During our knot-theory lecture we have talking about the following theorem: Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
3
votes
3answers
222 views

How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
4
votes
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 ...
5
votes
3answers
240 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
votes
3answers
965 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 ...
2
votes
2answers
878 views

Generalized Quaternion Group

Let $w = e^{\Large\frac{i\pi}{n}} \in \mathbb{C}.$ Prove that the matrices $X=\left( \begin{array}{cc} w & 0 \\ 0 & \overline{w} \\ \end{array} \right)$ and $Y = \left( \begin{...
3
votes
1answer
689 views

What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or ...
3
votes
1answer
137 views

Peculiar presentation of Symmetric group of degree 10

In the context of some problem I am working on, I got this peculiar presentation of a group. I have established computationally that this group is $S_{10}$, but I was wondering if it can be done ...
14
votes
3answers
2k views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
0
votes
1answer
97 views

If $a^{3^3}=b^9=1$ for generators $a,b$ of $G$, can we conclude that $G$ is a $3$-group?

I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$. Are these informations sufficient to affirm that the group is a $3$-group? Adding the relation $b^{-1}ab=a^4$, can we state ...
5
votes
0answers
291 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 ...
2
votes
2answers
193 views

Unusual presentation of a cyclic group.

Show that the group with presentation $$\langle a,b| aba^{-1}=b^n, b=(ba)^2\rangle$$ is a cyclic group generated by $a$ and determine its order.
4
votes
0answers
97 views

Showing $H=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k\rangle$.

Let $G=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k \rangle$. Prove that $G$ can be generated with $ab$ and $ab^{-1}ab$. And from there, $\langle(ab)^n\rangle\subset Z(G)$. Problem wants $H=\langle ab,...
9
votes
3answers
388 views

Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?

All is in the title: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$? I think it is the case, but I don't know how to prove it.
1
vote
2answers
329 views

Is there a residually finite group not finitely presented?

I am looking for a residually finite group which is not finitely presented. Does such a group exist?
2
votes
1answer
88 views

Central divisible subgroup

Have you any nice example of central divisible subgroup of a finitely presented group ? (of course the subgroup has not to be trivial)
23
votes
1answer
3k views

Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
7
votes
2answers
1k views

Presentations for alternating groups

Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators $a_n=(123),b_n=(1,2,\ldots,n)$?
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= \...
2
votes
2answers
259 views

$\langle X|\emptyset\rangle\ncong\langle X|R\rangle$ for finitely presented groups (exercise in Massey)

Let $:F_n$ denote the free group of rank $n$. How can I solve the exercise 7.6.3.(b), page 234, in Massey's Algebraic Topology? I'm guessing there's been made a mistake and (b) actually reads "If $G$ ...