Questions tagged [combinatorial-group-theory]

Use this tag for questions about free groups and presentations of a group by generators and relations.

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How to compute $Z(G)$, $G'$ and $G/Z(G)$ when $G$ is written in terms of generators and relations [closed]

Let $r,s,t\geq 1$, $1\leq i\leq r$ and $p$ prime. Consider the group $ G_i =‎\langle ‎a,a_1 ‎,a_2 ‎,‎‎\ldots ‎,a_{2s}|a^{p^{r+t}}=1, ‎a_1^{p^r}‎=‎\cdots ‎=a_{2s}^{p^r}=a^{p^i}, ‎[a_1 ,a_2 ]=[a_2 ,a_3 ...
Mahtab's user avatar
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1 vote
1 answer
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Explicitly finding a finite abelian group from a presentation, $\langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$

I have the following group presentation: $ G= \langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle $ It is clear that $G$ is a finite abelian group. I am interested in knowing what exactly $G$ ...
eyp's user avatar
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1 vote
1 answer
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In a finitely presented group, the set of all words which equal 1 in the group is a recursively enumerable set.

I was reading "An introduction to the theory of groups" by Rotman, chapter 12, the word problem. I am stuck in the following theorem, Let $G$ be a finitely presented group with presentation $...
Dwaipayan Sharma's user avatar
3 votes
2 answers
76 views

Subgroup of braid group $B_3$ isomorphic to itself

Consider the braid group $$B_3=\langle x,y:xyx=yxy\rangle.$$ It has a proper subgroup $N$, defined as follows: $g$ is in $N$ if and only if the sum of all exponents in $$g=\prod u_i^{\varepsilon_i},\ ...
atzlt's user avatar
  • 562
9 votes
2 answers
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Finding free subgroup $F_2$ in the free product $\frac{\mathbb{Z}}{5\mathbb{Z}} * \frac{\mathbb{Z}}{6\mathbb{Z}}$

Is there any free group isomorphic to $F_2$ contained in the free product group $\frac{\mathbb{Z}}{5 \mathbb{Z}}* \frac{\mathbb{Z}}{6 \mathbb{Z}}?$ Let $\frac{\mathbb{Z}}{5\mathbb{Z}}= \langle a \mid ...
jay sri krishna's user avatar
3 votes
0 answers
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Burnside groups with GAP system [closed]

My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP. The obvious representation using relations (see example for ...
arthurbesse's user avatar
0 votes
0 answers
34 views

Unsolvability of word problems in Groups

I was reading Rotman's "An Introduction to the theory of Groups" Chapter 12 "The Word Problem". I cannot understand the key concepts of Turing Machines and undecidibilty. I want to ...
Dwaipayan Sharma's user avatar
2 votes
1 answer
64 views

A generalization of Baumslag-Solitar groups

I am wondering about the following generalization of the group $B_{1,2}=\langle a,b\, |\, bab^{-1}=a^2\rangle$: $$ G_k=\langle a_1,a_2,\ldots,a_{k+1}\, |\, a_{i+1}a_ia_{i+1}^{-1}=a_i^2, i=1,2,\ldots,k\...
QMath's user avatar
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2 answers
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Using combinational group theoretical perspective on semidirect products, show $\langle r,s\mid r^8, s^2, srs=r^3\rangle$ has two Klein four subgroups

Note: This is an alternative-proof question, since I know how to prove the result but I'm asking for a particular kind of proof. Why? For the fun of it! Motivation: I've been trying to give a reason ...
Shaun's user avatar
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-1 votes
3 answers
170 views

Non-cyclic subgroup of order 4 in non-dihedral group

A group $G$ has sixteen elements: $$\{e, r, r^2, \dots , r^7, s, rs, r^2s, \dots , r^7s\},$$ where $r$ and $s$ satisfy the relations $r^8 = e, s^2 = e, sr = r^3s$. (Note that $G$ is not a dihedral ...
mathussy's user avatar
1 vote
0 answers
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Are sequenceable groups self-reducible?

A non-trivial finite group $G$ of order $n$ is said to be sequenceable if its elements can be arranged in a sequence ($g_{1}, g_{2}, \dots ,g_{n}$) in such a way that the partial products ($a_{1}, a_{...
SUTANAY BHATTACHARJEE's user avatar
3 votes
0 answers
67 views

Dependence relations in the group $a^4=b^4=(ab)^{15}=1$

There are two generators: $a$ and $b$. It is known that $a^4 = b^4 = (ab)^{15} = (ab^2)^6 = (ab^3)^9 = 1$. Is it possible and how to express relations $(ab^2)^6 = (ab^3)^9 = 1$ through relations $a^4 =...
Loom's user avatar
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2 votes
1 answer
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Is there an enumeration of finitely presented groups?

I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a ...
Fernando Chu's user avatar
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2 votes
1 answer
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Find a relation in GAP

I have a finitely presented group $G$ and a finitely generated subgroup $H<G$. GAP computed that $H$ has finite index in $G$. However, PresentationSubgroupMtc(G,H) cannot compute the presentation ($...
QMath's user avatar
  • 427
2 votes
3 answers
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Is this a correct definition for generating set of a group?

The Wikipedia article doesn't define the criteria for a generating set for a group in terms of logical propositions, so here's my attempt at it. It seems from (Why do generating sets need not contain ...
Nate's user avatar
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0 answers
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How can I decide whether two groups defined by finite presentations are (or not) isomorphic?

I have the groups $G_1,G_2$ with presentations $$G_1 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{-2} \rangle = \langle x,y : x^{-1}y^2 x = y^{-2}, yx^2y^3 = x^{-2} \rangle \\ G_2 = \langle x,y : (...
Adrian's user avatar
  • 61
1 vote
1 answer
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Defining a group isomorphism by mapping generators to generators

I am trying to understand how a group homomorphism mapping generators to generators can be shown to be an isomorphism. The example I have in mind is $S_3$ and $D_3$. The group presentations I have in ...
Valor Vaporeon's user avatar
2 votes
1 answer
133 views

Ways to show that words with exponent sum zero for each generator are elements of the commutator subgroup

Say I have a free group on the generators $X = \{ x_1, x_2, ... , x_k \}$ with $k \geq 2$. I read (in an article) that if I have a word $w$ written in the generators and their inverses, and the ...
Andreas Faltin's user avatar
1 vote
2 answers
140 views

Group of order $22$ is generated by two elements

I am trying to solve the below problem. I think this problem has been addressed on this website before, but I'd really like to work through the details myself without reading a prepared solution. Let ...
Valor Vaporeon's user avatar
4 votes
0 answers
74 views

Tietze transformations and the translation of words

I am new to combinatorial group theory and have a question regarding Tietze transformations. Say I have a group G, finitely presented by $\langle X | R \rangle$, and a sequence of (finitely many) ...
Andreas Faltin's user avatar
0 votes
0 answers
76 views

Find the order of the group from its presentation?

In defining a group presentation, it is natural to mod out by the subgroup generated by the 'relators', but there is a technical difficulty, this subgroup is not necessarily normal. So we can define ...
NotaChoice's user avatar
-1 votes
1 answer
65 views

Can you determine the order of a generator in this group presentation? [closed]

Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x. My follow up question: Is there a way to determine the order without finding ...
Björn's user avatar
  • 140
2 votes
1 answer
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Schreier basis of kernel of $F(G)\to G$ for $G$ a group

Let $G$ be a group and $F(G)$ the free group on $G$ as a set. There is a natural epimorphism $F(G)\to G$ that maps $[\sigma]\in F$ to $\sigma$, let $K$ be its kernel. Is the set $$X=\{[\sigma][\tau][\...
Hilbert Jr.'s user avatar
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2 votes
1 answer
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¿Tree of representatives contains exactly one vertex of each orbit?

I'm reading the book "Trees" by Serre and I'm having doubts about my understanding of the following statement. In page $25$ it says the following "A tree of representatives of $X$ mod $...
Benjita's user avatar
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3 votes
0 answers
97 views

Elements that satisfy an equation in an affine Artin group of type $\widetilde{A}_2.$

In order to properly express my question, lets set the notation. Consider the group $$A[\widetilde{A}_2]:=\langle a,b,c\ |\ aba=bab,aca=cac,bcb=cbc\rangle,$$ that is, an Artin group of type $\...
José Gálvez Mateos's user avatar
2 votes
0 answers
99 views

Why is this simplified proof of the Adian-Rabin theorem incorrect?

The proof of the Adian-Rabin theorem involves constructing a class of groups $W_w$ for a word $w$ in some group $U$ with undecidable word problem, and showing that this group either has some Markov ...
Perry Bleiberg's user avatar
3 votes
1 answer
110 views

$o(G) \leq o(a)o(b)$?

Let $G$ be a group such that $G = \langle a, b \rangle$. Mainly I wanted to see if $o(G) = \text{lcm}\left(o(a), o(b)\right)$, but I know $D_6$ is a counterexample. But I can't think of any ...
user avatar
3 votes
2 answers
101 views

Free Group where generators aren't just one symbol?

The question is: Show that $G = \langle a, b, c \mid a^2bacacab \rangle$ is a free group on the free generators $ab$ and $ac$. This is in section 1.4, page 39 of Combinatorial Group Theory by Magnus, ...
N A's user avatar
  • 51
2 votes
1 answer
67 views

Finite index subgroups of amalgamated free products over finite index subgroups

Let $G = H_1 \ast_K H_2$ be an amalgamated free product of two groups such that $K$ has finite index in both $H_1$ and $H_2$. Let $G'$ be a finite index subgroup of $G$. Does it follow that $G'$ ...
Jean Charles's user avatar
1 vote
0 answers
56 views

Dehn presentation implies finitely many conjugacy classes of elements of finite order

Let $G$ be a finitely generated hyperbolic group. Show that $G$ contains only finitely many conjugacy classes of elements of finite order. In “Geometric Group Theory: An Introduction” by Clara Löh, it ...
cede's user avatar
  • 613
2 votes
0 answers
55 views

Presentation of Product Group

Here is the question I have been working on: If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$. Deduce that, if $G_1$ and $G_2$ are ...
Happy Manager's user avatar
9 votes
1 answer
349 views

Can you completely determine a finitely presented finite group?

Let $G = \langle S \mid R \rangle$ be a finitely presented group. Suppose you know $G$ is finite. Can you completely construct the group multiplication table? I feel like the answer is yes. My first ...
cede's user avatar
  • 613
1 vote
1 answer
97 views

Extensions of alternating group by cyclic group

For integers $n\geq5$ and $n\neq6$, and $m\geq2$, by an extension $G$ of alternating group $A_n$ by cyclic group $\mathbb{Z}_m$, I mean the following short exact sequence: $1 \to A_n \to G \to \mathbb{...
Rajesh Dey's user avatar
1 vote
0 answers
63 views

Trying to find the set of unique representatives for the geodesics in the group $\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$

I am studying the conjugacy growth of the groups, and I encountered the following group: $$G=\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$$ Thanks to Derek for pointing out that $G$ is an ...
ghc1997's user avatar
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1 vote
0 answers
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$\text{GL }_2(\mathbb{Z})$ as an HNN extension

This question arises from section $2$, exercise $13.8$ of Bogopolski's Introduction to Group Theory. I managed to show that $\text{GL}_2(\mathbb{Z})\cong D_4 *_{D_2} D_6$. Now, I want to take a random ...
defacto's user avatar
  • 623
5 votes
0 answers
109 views

Number of groups with a bounded short presentation

How many groups there are (up to isomorphism) with a presentation with at most $n$ generators and with relators of length at most $3$? I don't expect there exist a sharp solution, since I know that ...
Dinisaur's user avatar
  • 1,055
4 votes
0 answers
109 views

Proving certain triangle groups are infinite

[Cross-posted to MathOverflow] Consider the Von Dyck group $$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$ where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family ...
Steve D's user avatar
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2 votes
1 answer
79 views

Prove that the group $G=\langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1\rangle$ is isomorphic to $S_4$

Prove that the group $$ G = \langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1 \rangle $$ is isomorphic to $S_4$. Obviously, take $a=(12),\, b=(23),\, c=(34)$ and use Von Dyck's Theorem ...
QiFeng233's user avatar
0 votes
0 answers
64 views

What is the group $G=\langle a,b \mid ab^2=b^3a,ba^3=a^2b\rangle $ [duplicate]

I feel that $G$ is infinite, however I don't know how to deal with the relationship $ab^2=b^3a,ba^3=a^2b$
QiFeng233's user avatar
14 votes
2 answers
616 views

Show a free group has no relations directly from the universal property

The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \...
cede's user avatar
  • 613
2 votes
0 answers
35 views

A generalization of Hurwitz group

Ignoring its geometric origin, a Hurwitz group might be defined abstractly by a non-trivial finite group which can be generated by elements $x$ and $y$ such that $x^2=y^3=(xy)^7=1$. I am wondering if ...
User0829's user avatar
  • 1,359
0 votes
0 answers
63 views

Converting any array of numbers to values between 1 and 100

I have several arrays of numbers. I need to convert each number in them to a value between 1 and 100. The lowest value in each array should be 1. The highest value in each array should be 100. I can ...
Bill Graham's user avatar
1 vote
0 answers
117 views

Is the isomorphism problem solvable for Euclidean groups?

Suppose you had two group presentations, and you know they are Euclidean groups, can you tell if they are isomorphic or not? It has been suggested to me that it is probably possible to tell if they ...
cede's user avatar
  • 613
2 votes
0 answers
73 views

What are some presentations of $SL(2,q)$?

In Presentation of SL$(n,\mathbb{Z}_p)$, it is asked whether there are known presentations of $SL(n,p)$. Its comments (particularly this one) and current answer hint at the existence of such ...
Shaun's user avatar
  • 45.2k
1 vote
0 answers
64 views

Help to understand the geodesics in $BS(1, 2)$

I would like to understand the sets of geodesics in $BS(1, 2)$, which is described in https://arxiv.org/pdf/1908.05321.pdf, Proposition 3 (page 3). Write $$ G=B S(1, 2)=\left\langle a, t \mid t a t^{...
ghc1997's user avatar
  • 1,431
1 vote
0 answers
58 views

A combinatorial property for finite cyclic groups

I was reading about the Cauchy–Davenport inequality and other results in combinatorics and the following property came to my mind. Let $G$ be a finite group. I call here $S=\{s_1,\ldots,s_n\}\subseteq ...
Alex Doe's user avatar
  • 634
2 votes
2 answers
179 views

Is this a valid group structure (of order 12)?

Let $$ G = \big\langle \, a, b, c \colon a^3 = b^2 = c^2 = (ab)^2 = (bc)^2 = 1, ac = ca \big\rangle. \tag{0} $$ That is, $G$ is a group that has elements $a$, $b$, and $c$ (though these are not the ...
Saaqib Mahmood's user avatar
0 votes
0 answers
72 views

linear isoperimetric inequality implies hyperbolicity

I am trying to find a nice proof that a finitely presented group satisfying a linear isoperimetric inequality implies it is hyperbolic. I came across these lecture notes, Theorem 3.22, but I am having ...
cede's user avatar
  • 613
3 votes
1 answer
93 views

Imre Ruzsa Generalisation of Kneser's theorem proof

According to this post here, there is a theorem by Ruzsa (2009) that for any group $G$ and $A,B\subset G$, $|A+B| \ge \min\{ p(G), |A|+|B|-1\}$ where $p(G)$ is the size of the smallest subgroup of $G$....
settheory's user avatar
6 votes
3 answers
245 views

Prove that $S_4$ is isomorphic to a presentation

I would like to prove that $G=\langle a,b \, | \, a^2,b^4,(ab)^3\rangle \cong S_4$. I tried to list out all the elements in the group presentation and show that it is isomorphic to $S_4$, but it was ...
Irene's user avatar
  • 521

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