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

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3
votes
1answer
37 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 ...
0
votes
0answers
42 views

Showing that a non trivial group 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
37 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 ...
1
vote
2answers
41 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
36 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
34 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
35 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 ...
0
votes
0answers
48 views

groups of order $p^4$

I need the classification of finite non-abelian groups of order $p^4$ from E. Schenkman's book "Group theory, 1965". Unfortunately our library has no this book and there does not exist the full ...
1
vote
1answer
27 views

A generating set of a finitely generated group

A group is called finitely generated if it has a presentation with finite generators. Edit: My original question was vacuous. Suppose that $G$ is a finitely generated group and $\{g_i\}_{i\in I}$ is ...
1
vote
0answers
47 views

Seminar topic in Combinatorial Group Theory

I am doing a course on "Combinatorial Group Theory" and we have a choice of giving a presentation on topics related to the course, Course outline is as follows- Free groups: groups defined by ...
1
vote
2answers
84 views

Semi group presentation $<a, b | a^{2} = b^{2} = 0, aba = a, bab = b>$

Another semi group question here, trying to get my head around the topic. Consider the semi group $S=\left<a, b | a^{2} = b^{2} = 0, aba = a, bab = b\right>$ I need to prove that $S$ has order ...
1
vote
1answer
17 views

Presentation of $Dih_n$

Let $\varphi:Z_2\rightarrow Aut(Z_n)$ be a homomorphism such that $\varphi(\overline{1})$ is the automorphism by inversion. Set $Dih_n\triangleq Z_n\rtimes_{\varphi} Z_2$. How do I prove that ...
3
votes
2answers
45 views

Minimal presentations and (co)homology groups

I wonder whether there exists a link between the number of generators and relations of a presentation for a given group $G$ and the ranks of its (co)homology groups $H_1(G,\mathbb{Z})$ and $H_2(G, ...
3
votes
1answer
111 views

Show that the group is trivial. [duplicate]

Show that the following group is identity: $$G=\langle x,y,z \mid xyx^{-1}=y^{2}\, , \, yzy^{-1}=z^{2}\, , \, zxz^{-1}=x^{2} \rangle.$$ This group is its own derived group. So all I get is group ...
8
votes
7answers
1k views

Why is this group not free?

$G= \langle x,y\mid x^2=y^2\rangle $. I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
0
votes
0answers
54 views

On the group algebra of a group specified by generators and relations

Let $k$ be a field, $G = \langle S \rangle/N(R)$ be a group specified by a set of generators $S$ and a set of relations $R$ (the brackets denote the free group, and $N(R)$ means the conjugate closure ...
2
votes
0answers
35 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 ...
2
votes
1answer
13 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 ...
5
votes
1answer
97 views

Presentation of a group isomorphic to $A_4$

I have a group $G$ defined by $G = \langle x,y,z|x^2 = y^3 = z^3 = xyz \rangle$ and we know that $a$ $=$ $xyz$ belongs to the centre of $G$. But im struggling to show that $\frac{G}{\langle a\rangle} ...
2
votes
1answer
71 views

centre of a group presentation

having trouble showing that an element belongs to a centre of a group presentation. Let $G = \langle x,y,z\mid x^2=y^3=z^3=xyz\rangle$ I have to show that $ a = xyz$ belongs to the centre of $G$. I ...
1
vote
0answers
28 views

Definition of minimal presentation of a group

I'm working on a problem on the braid monodromy of complex lines arrangements in $\mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere. Let ...
0
votes
0answers
40 views

How can I work out if a certain group presentation implies a certain relation?

I thought that maybe it would be possible to answer this question using the concept of a group presentation. Let $x_1,x_2,\ldots,x_k$ be k different elements of a group G and $k\geq4$. If we ...
3
votes
2answers
45 views

The group $\langle a,b,c \ | a^3,b^2, ab=ba^2, c^2, ac=ca, bc=cb \rangle$?

The group $\langle a,b,c \ | a^3,b^2, ab=ba^2, c^2, ac=ca, bc=cb \rangle$ is isomorphic to which permutation group. I have calculated its order and it is $12$, so my guess was $A_4$ but it is not ...
3
votes
0answers
40 views

Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, ...
2
votes
0answers
72 views

Trying to find a group isomorphism

I'm trying to find an isomorphism of a group with the following presentation:$$\langle a,b \mid (ab)^2=(abaa)^2=(abbb)^2=e\rangle$$ Basically, I'm not that experienced with groups so I'm wondering if ...
3
votes
1answer
84 views

Presentation of a non-abelian group of order $p^4$ such that ${G}/{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$

Let $G$ be a finite non-abelian $p$-group of order $p^4$ and $\frac{G}{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$, where $p$ is a prime. What is the presentation(s) of $G$?(If $G$ exixsts). Thanks ...
6
votes
1answer
85 views

on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
1
vote
2answers
63 views

Trying to understand group presentations using the example of the Dihedral group

According to Wikipedia the Dihedral group $D_n \cong \; \langle r,s \mid r^n = 1, s^2 = 1, s^{-1}rs = r^{-1}\rangle$. But why does this apply? As far as I understand the group presentation means that ...
8
votes
2answers
291 views

Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group. I have tried ...
0
votes
1answer
43 views

Prove that $Q_8 \cong \langle a, b \mid a^4, a^2b^{-2}, aba^{-1}b \rangle$

Prove that if $G = \langle a, b \mid a^4, a^2b^{-2}, aba^{-1}b \rangle$, then $G \cong Q_8$. I started by trying to define a homomorphism $\varphi: F(a,b) \to Q_8$ by $\varphi(a) = i$, $\varphi(b) = ...
0
votes
1answer
119 views

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can't find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the ...
1
vote
0answers
50 views

What makes $B_4$ different from other braid groups?

I am reading Braid Groups by Christian Kassel and Vladimir Turaev (GTM 247). Exercise 1.1.6 of the book is: Prove that each element $\sigma_i\sigma_j^{-1}$ with $1\leq i<j\leq n-1$ belongs to ...
1
vote
2answers
58 views

Normal Form of Elements in Quotient Groups

Let $G=⟨ S\mid R_1⟩$ be a group, where $S$ is the set of generators and $R_1$ is the set of relations. Let $H=⟨S\mid R_1, R_2⟩$ be the quotient group $G$ obtained from $G$ by adding a (possibly ...
3
votes
1answer
78 views

Forgetting a Strand in Braid Groups

Let $B_n$ be the braid group of $n$ strings over the unit disk $D$. Let $$d_i:B_n\to B_{n-1}$$ be the operation which is obtained by forgetting the $i$-th strand, $1\leq i\leq n$. Geometrically this ...
1
vote
1answer
40 views

Conditions for a finitely generated group with finite ordered generators

What are the conditions for a finitely generated group $G$ with finite ordered generators say $a_1, a_2,...,a_n$ to be finite? Note:I know that if $G$ is abelian, then it is finite. Are there any ...
3
votes
2answers
96 views

Presentations representing different groups.

Using GAP, I knew that groups $G=\langle x,y;x^4,x^2y^2,xyxy^{-1}\rangle$ and $H=\langle x,y;x^4,y^4,xyxy^{-1}\rangle$ are different. But I want to prove it. I tried to do something using Tietze ...
2
votes
1answer
85 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 ...
4
votes
1answer
72 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
2
votes
2answers
56 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 ...
6
votes
1answer
67 views

Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
4
votes
2answers
239 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
6
votes
2answers
103 views

Explicit description for $G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle$

I am trying to give an explicit description of the group $$G=\langle a,b,c\mid[a,b]=b\,,\,[b,c]=c\,,\,[c,a]=a\rangle\,.$$ Generalizing to fewer generators, one ends up with the trivial group, i.e. ...
1
vote
2answers
68 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 ...
3
votes
2answers
70 views

A finitely presented group $G$ is given by a presentation $\langle S,R\rangle$, where $R$ is finite. Show that $S$ is finite.

We were asked to prove this theorem in an exercise. This is what I have thus far: Suppose $S$ were infinite. Denote the set of symbols of $S$ that do not occur in any relation by $S'$. Then the free ...
1
vote
1answer
70 views

Show that $\langle a,b | ababa \rangle $ is a presentation of $\mathbb{Z}$.

I have to solve the following exercise: $\langle a,b | ababa \rangle $ is a presentation of $\mathbb{Z}$. Hint: Let $t =ab$ How can you I show such a thing? Help would be very appreciated.
6
votes
2answers
252 views

Showing that a group with a presentation is free/not free

Show that the group with presentation $\langle a, b, c \mid a^2cb^3\rangle$ is free with basis $\{a, b \}$. Show that the group with presentation $\langle a, b, c \mid a^3b^3 \rangle$ is not free. ...
6
votes
1answer
69 views

Finitely Presented is Preserved by Extension

Given $N= \langle n_i|r_j \rangle$ and $G/N= \langle g_k|s_l \rangle$, how do we prove $G$ has a finite presentation? We know that $G$ is f.g. by $\{n_i,g_k\}$ (I am being sloppy about directly ...
0
votes
0answers
41 views

Dehn presentation question

I have just shown that if a group $G$ admits a Dehn presentation then there are finitely many conjugacy classes of finite order. I'm then trying to deduce from that fact that there is some $N$ such ...
2
votes
1answer
89 views

Examples of non-finitely presented groups

I know several constructions leading to finitely generated non-finitely presented groups, using amalgamated products: Property: Let $A,B$ be two finitely presented groups. Then $A ...
2
votes
1answer
50 views

show trivial group

I have a group presentation here, which is $\langle a,b|a^n = b^{n+1}, aba=bab\rangle$, $n$ is any fixed integer. I want to show this presentation is in fact the trivial group. So far I have been ...