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

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1answer
34 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
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1answer
35 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
76 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 ...
3
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1answer
31 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
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1answer
24 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
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2answers
91 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
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2answers
55 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
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1answer
74 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 ...
6
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1answer
82 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
34 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
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0answers
41 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 ...
0
votes
1answer
49 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 ...
2
votes
1answer
60 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
52 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
67 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 ...
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0answers
45 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
39 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
52 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
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0answers
54 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
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1answer
40 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
37 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 ...
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0answers
53 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
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1answer
30 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 ...
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0answers
57 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
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2answers
88 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
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1answer
18 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
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2answers
53 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
122 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 ...
11
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8answers
2k views

Why the group $\langle x,y\mid x^2=y^2\rangle $ is 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
55 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
1answer
63 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
38 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
104 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
72 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
39 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
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0answers
44 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
56 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
42 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
77 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
87 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
93 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 $\langle a,b,c\mid a^2b^2c^2\rangle$?
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2answers
68 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
320 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
44 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) = ...
1
vote
1answer
175 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 ...
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0answers
54 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
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2answers
60 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
79 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
48 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
101 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 ...