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Questions tagged [group-presentation]

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

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1answer
164 views

Finite presentation of a cyclic group

I'm struggling to understand group presentation. There is a theorem that says, every group $G$ is the image of a suitable free group F (free upon a set $X$), and there must exist a homomorphism $\pi$ ...
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1answer
213 views

Isomorphism between groups

Suppose I have $2$ finite groups $G,H$ that have the same presentation. This means: $$G = \langle a_1,a_2, ..., a_n \mid \text{several conditions on these elements} \rangle$$ $$H = \langle b_1,b_2, .....
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1answer
360 views

Prove every free group is torsion free

Let $F$ be a free group and let $a \in F$ be an element of finite order, i.e. $a^n=1$ for some $n$. Also $a=a_1...a_s$ is a reduced word of length $s>0$, i.e. $a$ is an non-identity element. It ...
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1answer
148 views

Group Presentation - Categorical Interpretation

In this post, user Martin Brandenburg wrote $$\langle x_1,x_2,\dotsc \mid R_1,R_2,\dotsc \rangle$$ is defined to be a group satisfying the following universal property: The group has elements $x_1,...
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2answers
95 views

Which group is this?

Define $G=\left\langle a,b\ |\ a^2=1, (ab^2)^2=1 \right\rangle $. This is an infinite group whose Cayley graph is best described as a two-dimensional grid. Is it a well-known group? What is known ...
2
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1answer
112 views

Higman's group is not trivial.

I'm trying to prove that Higman's group is not trivial. In order to do that, first of all I have to define the following groups: $ \langle h_{i},h_{i+1}| h_{i+1}h_{i}h_{i+1}^{-1}=h_{i}^2\rangle$ for ...
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0answers
702 views

computing the fundamental domain for $\Gamma_0(4) \backslash \mathbb{H}$

how do I compute the fundmental domain for a congruence subgroup of $SL(2, \mathbb{Z})$ This region is important because the theta function $\theta(z) = q^{n^2}$ is invariant under two ...
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1answer
75 views

Confusing notation: Presentation theory of symmetric groups (Coxeter presentations)

I am having trouble to understand the notations used in Coxeter presentations of symmetric groups. The notation is defined as follows. Suppose $G$ is the symmetric group on the set $\left\{1, 2, 3, \...
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1answer
30 views

Reducing words in a set of relators

I have a set of relators $[r_1, r_2, r_3, r_4]$ of a free group on two generators $a$ and $b$, where $\begin{cases} r_1 = a^{2p} \text{ (where $p$ is some positive integer) } \\ r_2 = b^4 \\ t_3 = (...
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0answers
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Naming $V_{8n}=⟨a, b|a^{2n}=b^{4}=e, ba=a^{−1}b^{−1}, b^{−1}a=a^{−1}b⟩.$

Introduction Many groups can be defined by certain group presentation for example the cyclic group , the group $\mathbb{Z}/ m\mathbb{Z}\times \mathbb{Z}/ n\mathbb{Z}$, the dihedral group $\ldots$ ...
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1answer
86 views

What is this presentation isomorphic to?

I am trying to determine the group given by the presentation: $\langle\ a, b, c\ \vert\ a^2=b^5,\ b^2=c^3,\ c^2=a^7\ \rangle$. I've been trying to tackle this problem by starting with the first ...
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0answers
77 views

Is $\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle$ a presentation of the free group on a single generator?

Is the following a presentation of the free group generated by a single element? $\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle.$ My thinking is the following: $abab = baba=b(bab)=b^2ab$ by ...
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1answer
90 views

Presentation of commutative group by integer matrix

The task states: "Find a integer matrix $A \in Mat_{3 \times 4} \mathbb Z$ which presents commutative group $\mathbb Z \times \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_4$ and find it's Smith ...
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2answers
46 views

Trouble with a group presentation

I'm trying to prove that the elements of the group $\langle h_{0},h_{1} | h_{1}h_{0}h_{1}^{-1}=h_{0}^{2}\rangle$ can be expressed uniquely as $h_{0}^{n}h_{1}^{m}$ for some $n$, $m\in \mathbb{Z}$. In ...
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1answer
233 views

Why is the binary dihedral group $BD_4$ the same as $\mathbb{Z}_4$?

I am trying to understand why the binary dihedral group $BD_{4m}, m \in \mathbb{Z}$, with presentation $\langle\ A, B \mid\ A^{2m} =1,\ A^m = B^2 = -1,\ BAB^{-1}=A \rangle$ is the same, when $m=1$, as ...
3
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1answer
74 views

Proof that finite symmetrized relator sets, which are $C'(1/6)$, with equal normal closures are unique

The following statement is made in the Wikipedia article on small cancellation theory without reference or proof. Can anyone either provide a proof or point me to a reference with a proof? The ...
2
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1answer
49 views

Presentation of $D_4 \rtimes Aut(D_4)$

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the presentation of $D_4 \rtimes Aut(D_4)$ is as follows. $$ D_4 \rtimes Aut(...
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0answers
120 views

Showing the Heisenberg group admits the presentation $\langle A, B, C\mid AC=CA, BC=CB, ABA^{-1}B^{-1}=C\rangle.$

Consider the subset of $SL_3(\Bbb Z)$ consisting of matrices of the form $$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$ for $a,b,c\in\Bbb Z$. ...
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2answers
240 views

Quotient of a finitely presented group by a finitely presented group

Is a quotient of a finitely presentable group $G$ by a subgroup $N$ necessarily finitely presentable? What about if the subgroup $N$ is also finitely presentable?
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1answer
322 views

The additive group of rationals is not finitely generated

The additive group Z of integers is generator by 1 but additive group of rational numbers is not why?
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1answer
113 views

How to convert one presentation into another? Please explain using a dihedral group as an example.

How can we convert a given presentation of a group $G$ into an another presentation? Would anyone please explain to me by converting two different presentations of a dihedral group? Thanks in ...
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1answer
298 views

Understanding group structure using GAP

My question is regarding how to understand structure of group using GAP. Using StructureDescription(G), GAP gives some information like it is direct product or ...
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2answers
189 views

Presentation of Dihedral Group

Consider the standard presentation of $D_{2n}$: $\langle r, s : r^n = s^2 =1, rs = sr^{-1}\rangle$. I have seen the latter relation given as $sr = r^{-1}s$ a few times. Is this correct, as well?
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0answers
42 views

Solvability of group presentations with 2 “almost disjoint” relations

I am interested in a certain type of $2$-relator group presentations arising in algebraic topology which have two relators that only contain a single generator in common. Specifically, suppose I have ...
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2answers
469 views

presentation of subgroup and quotient group.

If we have a presentation $\langle x_i: R_j\rangle$ of $G$, (1) what does the presentation (in terms of the given generators and the given relations) of a subgroup of $G$ looks like? In particular, ...
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0answers
31 views

Information on relations from cohomology

I found in Brown's book, Cohomology of groups, that if a group $G$ admits a presentation $\langle g_1, \ldots, g_n \mid r_1, \ldots, r_m \rangle$, then the relation $$\mathrm{rank}~H_2(G) \leq m-n+ \...
1
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1answer
95 views

Group Presentation, Identification

everyone, how is everything going? Can someone help me to identify the group $$\langle a,b| a^4=b^3=1, ba=a^3b \rangle$$ My guess is that this group can be identified bay $C_6$, but I am in trouble ...
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1answer
285 views

One relator groups shortest word in quotient

Suppose we have a group $\langle S \vert r \rangle \cong F_S /\langle \langle r \rangle \rangle$ where $S$ is a finite set of generators and $r \in F_{S}$, i.e. a finitely generated one realtor group. ...
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3answers
155 views

A group given with presentation is finite or infinite?

The following was an exercise to check if a group is trivial or finite. The group is given by $$G=\langle x,y : yxy^{-1}=x^2, xyx^{-1}=y^2\rangle.$$ Question: Is $G$ a trivial group? or finite group?...
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0answers
76 views

Factor group of a $p$-group

I asked this question a few days ago in MO but got no answer, so I try here. Any hint will be appreciated. Let $$M(p^3)=\langle a, b\mid a^{p^2}=b^p=1, a^b=a^{p+1}\rangle$$ and $$G=\langle a, b\...
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0answers
32 views

Techniques for finding a complete set of relations for a finitely presented group

I am working on a problem where I want to find a presentation of a particular finite group. I have a particular transitive free action I'm interested in, and using this, I've been able to essentially ...
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0answers
91 views

Commutative diagram of semidirect products

My question is partly motivated by trying to solve this one. Let $E$ be the semidirect product of groups $G$ and $H$. Then, we have an exact sequence: $$G \hookrightarrow^\iota G \rtimes_\phi H \...
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256 views

Presentations of Semidirect Product of Groups

I have seen here that given two groups $G=\langle X|R \rangle := F(X)/N(R)$ and $H=\langle Y|S \rangle := F(Y)/N(S)$, then their semidirect product can be written as: $$ G\rtimes_\phi H \;=\; \langle ...
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0answers
68 views

How to show $\langle x,y \mid x^3=y^3=(xy)^3=1\rangle$ is presentation of an infinite group? [duplicate]

How to show $\langle x,y \mid x^3=y^3=(xy)^3=1 \rangle$ is presentation of an infinite group? This is a result in textbook, but I do not understand the rationale.
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201 views

Normal and non-normal subgroups of a finite $p$-group [closed]

Any hint about the following two questions will be greatly appreciated! Let $G$ be a finite non-abelian $p$-group with the following presentation: $$G=\langle a, b\mid a^{p^n}=b^{p^m}=1, [a,b]=a^{p^{...
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1answer
84 views

Finitely presented monoid with non-solvable word problem

A. Markoff (On the impossibility of certain algorithms in the theory of associative systems) and E. Post (Recursive Unsolvability of a Problem of Thue) provide examples of a finitely presented monoid ...
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3answers
337 views

Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.

I'm trying to prove that the following group has infinite order: $$H=\langle a,b\mid a^{3}=b^{3}=(ab)^{3}=1\rangle.$$ Currently I'm checking on some cases using the relations, but my problem is the ...
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1answer
37 views

For a Group $G$ with$S\subseteq G$, when does $ \left\langle S \right\rangle $ equal the normal closure of $S$?

To obtain the group with the presentation that has all elements of a subset $S$ equal to the identity, you must take the quotient of the free group (on the appropriate number of generators) that sets ...
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0answers
101 views

How to write $\Bbb Z_q\rtimes\Bbb Z_p$ as $\langle a,b\mid a^p=b^q=1,aba^{-1}=b^{i_0}\rangle?$

I am referring specifically to this example http://planetmath.org/groupsoforderpq In Case 2, the group should be $\mathbb{Z}_q\rtimes\mathbb{Z}_p$. How do we write it in the presentation of $$G=\...
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1answer
37 views

If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
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2answers
221 views

How to write the commutator subgroup in terms of the generators of the group?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. The commutator subgroup of $G$ is the group generated by $\{[a,b]\ |\ a,b\in G\}$ and is denoted by $[G,G]$, where $[a,b]=aba^{-1}b^{-1}...
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1answer
142 views

Finitely presented subgroup of finitely presented group

If I am given a group $G$, which is finitely presented by $\langle S \mid R \rangle$, and I am given a finitely presented subgroup $H$ of $G$. Is it true that $H$ takes the form $\langle T \mid R' \...
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0answers
104 views

Finding presentations of (finite) metacyclic groups: Doubts about one of D. L. Johnson's books.

I am reading the book "Presentations of Groups", by D.L Johnson and I have doubts about page 88, Proposition 1. Why is there a need to prove that $N \cong \mathbb{Z}_m$? Because it's given that $...
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1answer
73 views

What are the conjugacy classes of the group $\{k,r \,| \, k^3=r^2=(kr)^4=1\}$?

How can I determine the conjugacy classes of a group if I have the presentation of the group? For example, we know that $S_4$ has the presentation $$ S_4=\{k,r \,| \, k^3=r^2=(kr)^4=1\}. $$ What are ...
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Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
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0answers
141 views

Symmetric group, Braid Groups, and related groups

The symmetric group, in terms of presentation, is given by a group with generators $x_1,x_2,\cdots,x_n$ with following types of relations: (R1) $x_1x_{i+1}x_i=x_{i+1}x_ix_{i+1}$ (R2) $x_ix_j=x_jx_i$...
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0answers
53 views

How can I identify the following group of three generators

Let us consider the group $G= \langle a,b,c \mid aba=bab, aca=cac \rangle$. How can I determine the group identical with this group?
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2answers
73 views

Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
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1answer
58 views

Von Dyck groups that are conjugated.

Let us consider the Von Dyck groups $$ D(a,b,c)=\langle x,y,z\mid x^{a}=y^{b}=z^{c}=xyz=1\rangle $$ and $$ D(a'.b',c')=\langle x,y,z\mid x^{a'}=y^{b'}=z^{c'}=xyz=1\rangle. $$ Suppose $$ \frac{1}{a}...
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0answers
73 views

Proving a group $G$ with presentation $\langle a,b\mid ab\rangle$ is isomorphic to $\Bbb Z$.

Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$ (as elements that generate the group). The ...