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

learn more… | top users | synonyms (1)

1
vote
1answer
30 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 $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
3
votes
1answer
46 views

On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
0
votes
2answers
34 views

Abstract finite group vs. finite group

What is the difference between the definitions of abstract finite group and finite group? I have some exposure to the finite group theory. But I came to know about abstract finite group from ...
5
votes
0answers
157 views

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...
0
votes
0answers
6 views

Presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$

I would like to determine the presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$. My effort: The presentation of the symmetric group $S_{N}$ is as follows. $\langle s_1, \ldots, s_{N-1} | (s_i ...
4
votes
1answer
269 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of ...
2
votes
1answer
41 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 ...
0
votes
1answer
24 views

Prove that the presentation of a cyclic group of infinite order is $\langle a\mid \rangle$

Essentially, i need this for a third year mathematics project and originally i thought i just needed to have something like this: The group with this presentation is explicitly realized by the set of ...
0
votes
1answer
75 views

Fundamental group of sphere with 2 handles with 2 mobius bands.

Is it real to calculate?? We have a sphere with with 2 handles and with 2 glued mobius bands (red on picture). So, i think we need to use Van Kampen Theorem 2 times. 1) $ X = X_1 \cup X_2 $ , ...
4
votes
2answers
47 views

Relations in Group Presentation

In an introduction to abstract algebra, I was recently introduced to the idea of presenting a group - minimally, a group is just a set of generators along with a set of relations amongst the ...
6
votes
2answers
74 views

Presentation of Groups

I have troubles to solve this kind of exercises. For example: Let $$G_1=\langle x,y |x^3=y^4=1\rangle,~~~G_2=\langle x,y |x^6=y^6=(xy)^3=1\rangle. $$ I want to check that $G_1$ is an infinite ...
3
votes
1answer
55 views

What is the most general group possible?

I have read that free groups are the "most general" groups given some generators (from Wikipedia): The construction of a free product is similar in spirit to the construction of a free group (the ...
4
votes
3answers
187 views

Getting the wrong order of a finitely presented group

Let $G=\langle x, y \mid x^4=y^3=1, y^{-1}xy=x^{-1}\rangle$. What is $G$? I started by taking $$y^2=y^{-1}xy y^{-1}x^{-1}= (y^{-1}xy) ...
3
votes
0answers
36 views

Presentation of the special linear group $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$

I'm trying to show that $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$. For this, I defined a homomorphism $f$ from $G$ into $SL_2(\mathbb{Z})$ by $$f(a)=\begin{bmatrix}0& ...
0
votes
1answer
45 views

Isomorphism between groups with same presentation

So say I have two groups $G$ and $H$. We let $G = \langle X \mid R\rangle$ and $H = \langle Y \mid R'\rangle$ where $|X| = |Y|$ and $R$ and $R'$ are the "same" relations. (For example if $(ab)^c = e$ ...
4
votes
4answers
92 views

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 ...
2
votes
1answer
36 views

Relation between $\langle X\cup Y|R\cup S\rangle$ and $\langle X|R\rangle,\langle Y|S\rangle$

Let $X$ and $Y$ be disjoint sets and $\langle X|R\rangle$ be the group given by the generators $X$ and relations $R$. Similarly for $\langle Y|S\rangle$. Is there a simple relation between $\langle ...
2
votes
1answer
25 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 : ...
0
votes
0answers
43 views

Unique homomorphism from a group defined by generators and relations to $S_3$.

Let $G = \langle a,b; a^2b^{-3} = 1 \rangle$ and consider $(1 2), (123) \in S_3$. I need to show there is a unique homomorphism from $G$ to $S_3$ which sends $a$ to $(12)$ and $b$ to $(123)$. I'm not ...
0
votes
1answer
46 views

Prove a group has a presentation (revisit)

I was actually asking the same question in here. However, the give answer didn't satisfy me. For the reader's convenience I'll re-write the post as follows: We define the following operation on the ...
0
votes
0answers
34 views

Which correct sentence to explain the function $g(\nabla I)=\frac{1}{1+\beta |\nabla(G_{\sigma}*I)|^2}$

I have a edge indicator function that has formula as $$g(\nabla I)=\frac{1}{1+\beta |\nabla(G_{\sigma}*I)|^2}$$ where $\nabla$ is gradient operator, $*$ is convolution operator, $G_{\sigma}$ is a ...
2
votes
1answer
57 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 $G/C$. ...
1
vote
0answers
33 views

Finite subgroups of $PSL(2,R)$

I know $PSL(2,R)$ is $SL(2,R)/SZ(2,R)$ and it is a simple group, but I do not have single clue how to get on finding its group presentation. How can I find its presentation and also I am looking for ...
1
vote
0answers
29 views

When is a finitely presented positively admissible group uniformly reachable?

This is a follow-up from this question. For a finitely presented group $G=⟨S|R⟩$, a positive element g∈G is an element of G that can be written as a finite product of elements of $S$ only. A ...
2
votes
3answers
97 views

Is a finite index subgroup of a finitely presented subgroup finitely presented?

I do know Schreier's theorem, which states that a finite index subgroup of a finitely generated group is finitely generated. Other than this, I have no reason to suspect a positive answer to my ...
2
votes
0answers
61 views

How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
5
votes
4answers
446 views

Does a Conjugacy Class always contain an element and its inverse?

The definition of a conjugate element We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$ Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = ...
6
votes
1answer
143 views

Is the minimum number of relations in a free product, the sum of the minimum number of relations in the free factors?

Say $\rho(G)$ is the minimum number of relations required to present the group $G$. Is $\rho(A*B)= \rho(A)+\rho(B)$? What can be said about $\rho(A*B)$? A while ago I was thinking about $C_3*C_4$, ...
2
votes
0answers
77 views

What is the group for the following presentation?

This group has four generators $a, b, c$ and $d$. Generators satisfy the following relations: $a$ commutes with $b$ and $c$ commutes with $d$. $a^2=b^2=1$ and $c^{m_1}=d^{m_2}=1$ for some integers ...
2
votes
1answer
49 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
4
votes
1answer
42 views

Presentations of the unity group

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for ...
5
votes
2answers
80 views

$G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free.

I have to prove that $G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free. Some things about this group that I understand are first show that $M=\langle (xy)^2,x^2,y^2 ...
4
votes
1answer
58 views

Presentation of groups and positive expressions

For a group $G=〈S|R〉$, $S,R$ are both finite. A positive element $g\in G$ is an element of $G$ that can be written as a finite product of elements of $S$ only. A positive expression of $g$ is a word ...
0
votes
1answer
34 views

Dihederal Group $D_{2n}$ Where $n$ is even/odd

I know that the group presentation of $D_{2n}$ is the following $$D_{2n} = \big<a,b: a^n=b^2=1,b^{-1}ab =a^{-1} \big>$$ Now if we consider the case where $n$ is even and we write $n =2m$ for ...
5
votes
2answers
110 views

How to obtain a presentation for each group of order $64$

I am new to his forum, and would like to know how to obtain a presentation for each group of order $64$. I wish to do this for all the groups of order $64$. Thanks in advance.
2
votes
2answers
83 views

Presentation of the additive group of the rational numbers

We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ...
2
votes
1answer
78 views

Prove $G \cong \langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$

Let $D^3= D \times D \times D$ where $D = D_\infty$ where we see $D$ as the group generated by $\mathbb{Z}$ and element $0^*$ of order $2$ such that $0^*n0^*=-n$ for all $n \in \mathbb{Z}$. Letting ...
7
votes
1answer
130 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
3
votes
1answer
37 views

simple question on conjugacy classes

if $ \;G = \langle a,b\;|\; a^9 = b^3 = 1, bab^{-1} = a^4\rangle\; $ of order $\;27\;$ Then how would i show that $b$ is conjugate to $ba^3$ I have been fiddling around with this for ages and cannot ...
2
votes
0answers
45 views

$X_{2n}$ be group presentation as displayed below proof verification

Hi so I am solving problems in dummit and foote, however this problem I am not able to do it Show that if $n = 3k$, then $X_{2n}$ has order 6, and it has same generators and relations as $D_6$ when x ...
1
vote
1answer
62 views

Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$.

I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look ...
3
votes
1answer
73 views

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 ...
3
votes
1answer
53 views

Prove that $g\in \ker \varphi$

Let $G \approx \langle a,b \mid b^{-1}a^3b = a^5 \rangle$. Let $H$ be a finite group and $\varphi:G \to H$ be a homomorphism. Then $g = a^{-1}b^{-1} a^{-1}bab^{-1}ab \in \ker \varphi$. My attempt: ...
3
votes
1answer
92 views

Working with finitely presented groups in GAP

This is more of a question specifically about how GAP handles calculations with finitely presented groups rather than about group theory. I have several finite group presentations that I would like ...
1
vote
1answer
122 views

Showing that a one-relator group, $\langle a,b \mid W \rangle$, is not a free product.

Let $W=W(a,b)$ be a cyclically reduced word in $\langle a,b;\emptyset\rangle$ not equal $1$ that contains at least one non-zero power of $a$ and at least one non-zero power of $b$. How can be proved ...
0
votes
1answer
45 views

How to find presentation of $G'$

If my $G=\langle a,b\ |\ a^4,b^4,a^2=b^2 \rangle$, then how to find presentation for $G'$ (derived subgroup of $G$). Should I proceed by Reidemeister-Schreier process, that is too cumbersome. Iisn't ...
2
votes
2answers
63 views

Equation over finite group with presentation $G=\langle a,b|R\rangle$

This question arised like a curiosity, I've been trying to find out information about the solution (after trying to solve it by myself) with no success. The question is: Given a finite group with ...
2
votes
0answers
73 views

Group presentation: How can we determine the group with the presentation below?

Please how can we determine the finite group whose presentation is given as: $$G:=\left\langle g,h|g^4=h^4=1,hg=g^{-1}h \right\rangle?$$
1
vote
1answer
47 views

Size of conjugacy classes in SL(2,3)

I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are: ...
2
votes
1answer
42 views

Notation of Burnside's group theory book “The theory of finite groups”

According to the classification of finite non-abelian groups of order $p^4$ in Burnside's book "The theory of finite groups, 1897, pages 87-88" one of the types of these groups is the following ...