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

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38 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 ...
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2answers
44 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
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1answer
68 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 ...
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1answer
30 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 ...
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2answers
90 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
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1answer
70 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
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1answer
65 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
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2answers
48 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
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1answer
62 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 ...
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2answers
211 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
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2answers
90 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. ...
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2answers
54 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
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2answers
62 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 ...
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1answer
66 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.
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2answers
236 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. ...
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0answers
35 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
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1answer
53 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
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1answer
46 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 ...
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2answers
77 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
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1answer
76 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. ...
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1answer
111 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
44 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
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4answers
424 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
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1answer
118 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
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2answers
106 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
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1answer
28 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 ...
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1answer
56 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
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2answers
141 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
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2answers
106 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? ...
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2answers
161 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
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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
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3answers
265 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
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3answers
98 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
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1answer
111 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
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3answers
165 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 ...
2
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1answer
58 views

How to show this presentation of the additive group $(\mathbb{Q},+)$?

The task is: Show that $$ \langle (x_n)_{ n \in \mathbb{N}} \mid x_n^n = x_{n-1} \text{ for } 1 < n \in \mathbb{N} \rangle $$ is presentation of additive group $(\mathbb{Q},+)$. Can you explain ...
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0answers
67 views

If $G/N$ and $N$ are finitely presented, then $G$ is finitely presented.

Let $G$ be a group and $N\triangleleft G$. Show that if $G/N$ and $N$ are finitely presented, then $G$ is finitely presented. Worked on this problem for about 2 hours before we all threw in the ...
2
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1answer
98 views

Dehn and Wirtinger Presentations of Knot Groups and their connection

I'm currently working through N.D. Gilbert and T. Porter's Knots and Surfaces. In it the idea of a Wirtinger presentation and a Dehn presentation for a group associated with a given knot is ...
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2answers
85 views

Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
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1answer
313 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
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0answers
92 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
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1answer
43 views

The deficiency of surface group

Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The ...
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0answers
37 views

The order of the group $\langle a, b| w^3, w\in\langle a, b\rangle\rangle$ for $w$ being any word.

I came across this group mentioned in passing as finite. Does anyone know the order of the group, and where I can find a proof of this quantity? Replacing $3$ with $n$, does this problem have a name ...
4
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1answer
94 views

Groups with a R.E. set of defining relations

Reading around I found the following two assertion: 1) Every countable abelian group has a recursively enumerable set of defining relations. 2) Every countable locally finite group has a recursively ...
5
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2answers
158 views

Finitely generated group which is not finitely presented

Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented? I was looking for a finitely presented group with a central ...
3
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0answers
167 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
4
votes
1answer
80 views

Showing a group is isomorphic to a group with known presentation

Let $H$ be a group with presentation $\langle h_1, \dots, h_n \mid r_1 = \dots = r_m = 1\rangle$. If there are $g_1, \dots, g_n \in G$ which satisfy the relations $r_1, \dots, r_m$, when is $\varphi : ...
5
votes
1answer
193 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 ...
3
votes
3answers
118 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 ...
1
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1answer
99 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 $|G|=12$.