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Questions tagged [group-presentation]

For questions concerning groups defined via a presentation by generators and relations. Should probably be used along with the general (group-theory) tag.

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Is the group $\langle x,y \mid xy^n = yx^n \rangle $ trivial?

I’m currently browsing through Clara Löh’s book “Geometric Group Theory - an introduction”, and came across the exercise 2.E.14, which gives some examples of presentations of groups and asks to ...
NonAffine's user avatar
0 votes
0 answers
43 views

Determining the fundamental group of the following quotient

Let $X$ be te quotient of a hexagon identifying the sides as in the scheme $aaabbb^{-1}$. By theorem 74.2 from Munkres I know that if all vertices are identified, then $$ \pi_1(X,x_0)\cong\frac{F(a,b)}...
Valere's user avatar
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8 votes
3 answers
328 views

Proving that a group is infinite and nonabelian

As an exercise I am trying to prove that the group $$G = \langle a,b,c \mid ac = ba, ab=ca, bc=ab\rangle$$ is infinite and non-abelian. Moreover, the author claims that its center has finite index. I ...
user1008978's user avatar
3 votes
1 answer
69 views

2 generated groups, with generators of order 3

I can show that if $G$ is a group generated by $a$ and $b$, with $a^2=b^2=1$, then $G$ is a finite elementary abelian 2 group or a dihedral group (finite or infinite), depending whether $ab$ has ...
user17488's user avatar
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3 votes
1 answer
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Prove a surjective endomorphism $\phi$ of a 1-relator group $ ⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩ $ is not injective

Consider the infinite group $H$ with presentation $$ ⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩ $$ so that the relation is $a^{-1} b^2 a=b^3$. The map $$ a ↦ a\\b ↦ b^2 $$ induces a surjective homomorphism $ϕ:H\to ...
hbghlyj's user avatar
  • 3,023
1 vote
1 answer
73 views

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
  • 127
3 votes
2 answers
88 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
-1 votes
1 answer
46 views

Given a group $G = \langle a,b,c\mid a^3, c^4 , c(ab)^2ca^{-1}\rangle$, show the existence of the homomorphism $\varphi : G \rightarrow S_6$ [closed]

Given that $G = \langle a,b,c\mid a^3, c^4 , c(ab)^2ca^{-1}\rangle$, show that there is a homomorphism $\varphi : G \rightarrow S_6$, such that $\varphi(a) = (1, 3, 5)(2, 4, 6),$ $\varphi(b) = (1, 2, ...
phyx's user avatar
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1 vote
0 answers
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metabelian groups can be represented by matrices, do we know the exact representation?

We know that every finitely generated metabelian group can be represented by matrices (e.g. see https://link.springer.com/article/10.1007/BF02219822 ). I am particularly interested in how finitely ...
ghc1997's user avatar
  • 1,511
2 votes
1 answer
71 views

Is there a good presentation of the matrix algebras?

Let $R$ be a commutative ring with identity. Suppose the matrix algebra $\operatorname{Mat}_n (R)$ and matrices $$E_{mk}:=(a_{ij})=\begin{cases} a_{ij}=1 \ \text{if} \ i=m \ \text{and} \ j=k \\ a_{ij}=...
Mitya Kustov's user avatar
2 votes
1 answer
66 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
  • 457
2 votes
2 answers
65 views

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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2 votes
3 answers
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Fundamental group of simplicial space

This question asks the same as mine, but it was unsuccessful in getting an answer, so I try again. For context I am reading Weibel's K-book and am struggling with proposition 8.4 which computes $K_0$...
DevVorb's user avatar
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2 votes
1 answer
76 views

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
  • 457
2 votes
1 answer
68 views

How does $ab= a^2ba$ for the Dihedral group $D_8 = \langle a, b \rangle$ and a $\mathbb{C}D_8$-Algebra?

I'm trying to show for $z = b + a^2b \in \mathbb{C}D_8$, $az = za$. I have the presentation $D_8 = \langle a, b \ : \ a^4 = 1, b^2 = 1, b^{-1}ab = a^{-1} \rangle$. So I want to show $az = ab + a^3b = ...
Wofster's user avatar
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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
4 votes
0 answers
81 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
78 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
0 answers
102 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
1 vote
1 answer
204 views

GAP Isomorphism between two finitely presented p-groups

I have finitely presented nilpotent p-groups with exponenta 5 G = $\langle x1, x2, x3, x4: [x1, x2] = 1, [x3, x4] = 1, [x1, x4] = [x3, x2], [x2, x4] = [x1, x3]^{-2}] \rangle$ and H = $\langle x1, x2, ...
Aiden Peterson's user avatar
2 votes
0 answers
107 views

Can the following proof calculus show that any finitely presented free group is free?

If a finitely presented group is free, will it always have a proof in the proof calculus outlined in this question that it is free? I recently saw this question. I tried to show that the group was ...
Greg Nisbet's user avatar
  • 11.8k
1 vote
0 answers
59 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
351 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
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
  • 1,511
1 vote
1 answer
40 views

presentation of a family of solvable groups

I am looking for a family of finite solvable groups (with infinite members) with given presentations. I am aware that for example dihedral groups, or dicyclic groups have well-known presentations. ...
Amin's user avatar
  • 700
1 vote
0 answers
47 views

$\text{GL }_2(\mathbb{Z})$ 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
  • 633
5 votes
0 answers
110 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
  • 1,085
1 vote
0 answers
53 views

Writing the product of cycles as disjoint cycles, but yielding to different result when applying it.

I'm studying the symmetric group $S_n$ of permutations from $\{1,2,\ldots,n\}$ to itself. I'm doing a problem where I have to show that $$S_4\cong \langle a,b~|~a^4=b^2=(ba)^3=1\rangle.$$ I thought ...
Fabrizio G's user avatar
  • 2,095
1 vote
0 answers
118 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
3 votes
1 answer
147 views

Another cell complex structure of closed orientable genus $g$ surface?

Suppose we have a wedge sum of $2g$ circles, labeled by $a_1,b_1,\dots, a_g,b_g$. It is well known that if we attach a 2-cell with attaching map given by $a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}...
blancket's user avatar
  • 1,760
2 votes
0 answers
77 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
  • 45.6k
1 vote
1 answer
48 views

Presentation of direct sum

I find that presentations are the most useful tool to get an intuitive feeling about the nature of some group construction. I have read here at mathstack that given $\{G_i = \langle X_i \mid R_i \...
Lucas Giraldi's user avatar
2 votes
0 answers
70 views

Word problem for groups and the universal cover

I am confused by certain constructions regarding fundamental groups. The word problem for finitely presented groups is known to be undecidable. However, it is also known how to construct, given a ...
roymend's user avatar
  • 436
2 votes
2 answers
181 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
6 votes
3 answers
267 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
0 votes
0 answers
48 views

Reference on group-presentations

I'm looking for an introduction to free abelian groups and their presentation, specifically to understand notation such as: $G=^{ab}\left\langle a,b,c \text{ } | a + 2b + 3c = −a − 2b − 7c = 0 \right\...
iki's user avatar
  • 181
1 vote
0 answers
80 views

Motivation for presentation of group $G$ in Hungerford’s Abstract Algebra

An immediate consequence of Corollary 9.3 and the First Isomorphism Theorem is that any group $G$ is isomorphic to a quotient group $F/N$, where $G =\langle X\rangle$, $F$ is the free group on $X$ and ...
user264745's user avatar
1 vote
1 answer
82 views

Compute the automorphism group of $D_{10}$ [duplicate]

This exercise will compute the automorphism group of $D_{10}$. The presentation of the group is $D_{10 }= <r, s |r^5 = s^2 = 1, s^{−1}rs = r^{−1}>$ (a) (i) Show that under any automorphism $σ : ...
ALMEra's user avatar
  • 357
0 votes
1 answer
85 views

Identify the presentation of groups [closed]

I'm trying to classify all groups of order $20$. I get two group presentation and want to identify isomorphism type. I think one of these isomorphic to $D_{10}$ and other isomorphic to $Dic_5$. Can ...
Alhabud's user avatar
  • 377
3 votes
1 answer
142 views

Show that $G = \langle a,b \mid a^2b = ba^3\rangle$ is nonabelian.

I was trying to suppose for a contradiction that it is abelian, then: $$ba^3 = a^2b = ba^2\implies a = 1$$ Then we have $G = \langle a,b\mid a = 1\rangle$, which I believe is isomorphic to $\mathbb{Z}$...
Paimonium's user avatar
9 votes
2 answers
523 views

Show, using presentations, that $\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,n)} .$

Note: This is an alternative-proof question and thus is not a duplicate. The Question: Show, using group presentations, that $$\Bbb Z_m\times\Bbb Z_n\cong\Bbb Z_{{\rm lcm}(m,n)}\times \Bbb Z_{\gcd(m,...
Shaun's user avatar
  • 45.6k
2 votes
1 answer
61 views

Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle $ is an inner automorphism

Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle $ is an inner automorphism. I found $Z(F_{20})={1}$ and it follows $F_{20} \cong \text{Inn}(F_{20})$. ...
Alhabud's user avatar
  • 377
13 votes
2 answers
198 views

Is $\Bbb Z^3$ a one-relator group?

I understand that: $\Bbb Z^0 = \langle a \mid a \rangle$ $\Bbb Z^1 = \langle a, b \mid b \rangle$ $\Bbb Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$ but is it possible for $\Bbb Z^3$ to be ...
cede's user avatar
  • 613
0 votes
0 answers
25 views

What are the elements of the Dihedral Group $D_8$ when given by $D_8 = \langle a, b \mid a^2 = b^2 = (ab)^4=1\rangle?$ [duplicate]

I'm very confused since I've always seen the Dihedral Group $D_8$ given as $$D_8 = \langle a, b \mid a^4 = b^2 =1, ab= a^3b\rangle.$$ I'm given the Definition $$D_8 = \langle a, b \mid a^2 = b^2 = (ab)...
tomato's user avatar
  • 9
2 votes
0 answers
123 views

When can we say $H$ is a subgroup of $G$ given their presentations?

Consider $F_2=\langle a,b \rangle$, the free group on $2$ letters. If I look at $H=\langle ab,ba^{-1}, bab^{-1} \rangle$ (the generators are arbitrary), since each generator can be made from $a$ and $...
Aditya's user avatar
  • 930
1 vote
1 answer
78 views

Presentation of Klein group: Show $G=\langle a,b\mid a^2,b^2,(ab)^2,(ba)^2,ab^2a\rangle$ is iso to $K_4=\langle x,y\mid x^2,y^2,xyx^{-1}y^{-1}\rangle$

Let $$G = \langle a, b \mid a^2, b^2, (ab)^2, (ba)^2, a b^2 a \rangle.$$ I need to show that this group is isomorphic to Klein Group $$K_4 = \langle x,y\mid x^2, y^2, xyx^{-1} y^{-1}\rangle.$$ I prove ...
Horned Sphere's user avatar
0 votes
0 answers
91 views

How to prove that a group affords no epimorphism in $S_3$

Good evening, I want to prove that the group $G = \langle a, d \mid a d a^{-1} d a = d a d^{-1} a d \rangle$ affords no epimorphism in $S_3$ (the group of the permutations of $\{1, 2, 3\}$). I tried ...
Arthur Filippi's user avatar
0 votes
1 answer
63 views

If $\forall n\in\Bbb N$, if $g\in G$ with $|g|=n$ implies $\forall m\in\Bbb N,\exists h_m\in G,|h_m|=n^m,$ then can non-torsion-free $G$ be f.p.?

Motivation and Details: Thinking back a couple of years ago, when I did combinatorial group theory a lot, I decided to explore the idea that whenever an element of a group has finite order $n$, then ...
Shaun's user avatar
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