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

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0answers
19 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
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1answer
12 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 ...
2
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1answer
74 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
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1answer
64 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 ...
0
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0answers
38 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
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2answers
42 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 ...
2
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0answers
31 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
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0answers
66 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
78 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 ...
4
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1answer
72 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
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2answers
59 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 ...
4
votes
0answers
42 views

Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $<x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2}>$ is equivalent to the trivial group. I have tried all sorts ...
0
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1answer
42 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
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1answer
93 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
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0answers
47 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
52 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
74 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
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1answer
39 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
92 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
78 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
53 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
230 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 ...
5
votes
2answers
96 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
62 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
66 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
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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
245 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
61 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
37 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
67 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
47 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 ...
4
votes
2answers
92 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
80 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
117 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
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1answer
47 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
435 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
120 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?
2
votes
2answers
125 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
30 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
57 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
144 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
117 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? ...
7
votes
2answers
196 views

Group presentations: What's in the kernel of $\phi$?

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
68 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
290 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
105 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
129 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
178 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 ...