Questions tagged [group-presentation]

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

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4
votes
1answer
99 views

Algorithmic way to check if a power-conjugate presentation is consistent?

Is there an algorithmic way to check if a power-conjugate presentation (for a finite polycyclic group) is consistent? Background: A finite solvable group $G$ has a subnormal series $$ G=G_0 \...
4
votes
1answer
111 views

Presentation of groups and positive expressions

For a group $G=〈S|R〉$, $S,R$ are both finite. A positive element $g\in G$ is an element of $G$ that can be written as a finite product of elements of $S$ only. A positive expression of $g$ is a word $...
4
votes
1answer
86 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 ...
4
votes
1answer
150 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 ...
4
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1answer
97 views

Is there an algorithm to solve all soluble group word problems?

What I mean is, is there an algorithm that given any finitely presented group with soluble word problem can solve the word problem on that group?
4
votes
1answer
175 views

Group Presentations and Cayley graphs

I am trying to understand group presentations and Cayley graphs, and have a few questions I am confused about. Let $G=(V,E)$ be a finite $d$-regular graph that is known to be a Cayley graph for the ...
4
votes
1answer
536 views

how to prove two groups are NOT isomorphic?

I have two groups defined by presentations $$\langle x, y \mid x^p = y^q \rangle$$ $$\langle x, y \mid x^{p'} = y^{q'} \rangle$$ where $p,q,p',q'$ are all integers greater then $1$, and $\gcd(p,...
4
votes
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 ...
4
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2answers
109 views

Relations in Group Presentation

In an introduction to abstract algebra, I was recently introduced to the idea of presenting a group - minimally, a group is just a set of generators along with a set of relations amongst the ...
4
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2answers
166 views

Minimal presentations and (co)homology groups

I wonder whether there exists a link between the number of generators and relations of a presentation for a given group $G$ and the ranks of its (co)homology groups $H_1(G,\mathbb{Z})$ and $H_2(G, \...
4
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1answer
188 views

Describing the group defined by $\langle a,b \mid (ab)^2=(abaa)^2=(abbb)^2=e\rangle.$

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 ...
4
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1answer
142 views

Identifying the group $G = \langle x,y \space | \space xy=yx, x^4=y^2 \rangle$ from the given presentation [duplicate]

I'm trying to solve a problem in my textbook which asks me to identify the groups $G_1 = \langle x,y \space | \space x^3y=y^2x^2=x^2y\rangle$ and $G_2 = \langle x,y \space | \space xy=yx, x^4=y^2 \...
4
votes
1answer
90 views

Showing $\langle a,b\mid abab^{-1}\rangle$ and $ \langle c,d \mid c^2d^2\rangle$ are isomorphic.

I computed the fundamental group of the Klein bottle in two different ways and obtained two seemingly different answers: $$ \langle a,b \mid abab^{-1}\rangle $$ and $$ \langle c,d \mid c^2d^2\rangle. $...
4
votes
1answer
115 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 ...
4
votes
1answer
97 views

Are these groups solvable?

I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype. We can think of them as follows: Start with an infinite cyclic group $\langle a\...
4
votes
1answer
191 views

Coxeter presentation of Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$.

I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation $$\langle s_{\text{1}},\ldots,s_n\mid s_{\text{1}}^{\text{2}}=s_i^2=1, (s_1s_2)^4=(s_is_{i+1})^3=(...
4
votes
1answer
56 views

Presentations of the unity group

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for ...
4
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1answer
471 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 introduced....
4
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1answer
187 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 : ...
4
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1answer
160 views

Determining $\langle a,b,c\mid 6a+9b+6c=0, 8a+12b+4c=0\rangle^{\operatorname{ab}}.$

I have to find out what is this abelian group (in the form $\mathbb{Z}/m_1\mathbb{Z} \times ... $). Its relations are: $$6a+9b+6c=0$$ $$8a+12b+4c=0$$ with generator $a,b,c$. My solution is: $$\mathbb{...
4
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0answers
61 views

$G_1$ a subquotient of $G_2$ and $G_2$ a subquotient of $G_1$, is $G_1 \cong G_2$?

Sorry for a presumably noobish group theory question. I would like an example of the following: $G_1,G_2$ are finitely presented groups such that there exists finitely presented groups $C_1,C_2$ ...
4
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0answers
52 views

Prove that finite and infinite presentations of Thompson group $F$ are isomorphic.

Let $$ G=\langle x_0,x_1,\dots\mid x_jx_i=x_ix_{j+1}\text{ for }i<j\rangle, $$ $$ H=\langle a,b\mid [ab^{-1},a^{-1}ba],[ab^{-1},a^{-2}ba^2]\rangle, $$ where $[x,y]$ is commutator. These are both ...
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0answers
96 views

Isomorphy of simple groups of order 360 : a proof with a presentation

It is well known that all simple groups of order 360 are isomorphic with the alternating group $A_{6}$. Cole's original proof is here on StackExchange : $A_6 \simeq \mathrm{PSL}_2(\mathbb{F}_9) $ ...
4
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0answers
85 views

Show that a group is trivial [duplicate]

Let $G$ be the group generated by $a,b,c$ with relations $ab=b^2a$, $bc=c^2b$, and $ca=a^2c$. Show that $G$ is trivial. A related problem is that for $G_1$ generated by $a,b,c,d$ with similar ...
4
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0answers
208 views

$G_n$'s mutually non-isomorphic

This question was answered by @Jim Belk And he defined $G_n$ as follows: $$ G_n \;=\; \langle a,b \mid [a^{-1}ba,b] = \cdots = [a^{-n}ba^n,b]=1\rangle $$ My question is: Why $G_n$'s are mutually non-...
4
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0answers
782 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 ...
4
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0answers
44 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 ...
4
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0answers
53 views

Construction of a group where nonempty words of small length are not equal to the identity element

Let $n$ and $p$ be positive integers. Is there a finite group $G_p$ generated by elements $a_1, \dots, a_n$ such that any nonempty reduced word on $a_1, \dots, a_n, a_1^{-1}, \dots, a_n^{-1}$ of size $...
4
votes
1answer
88 views

Endomorphism of a group

Let $G$ be a group with presentation $$G=\langle x_1,x_2,\cdots,x_k\colon R_1(x_1,\cdots,x_k)=1, \cdots, R_n(x_1,\cdots,x_k)=1\rangle.$$ Here $R_i(x_1,\cdots,x_k)$ denotes a word in $x_1,\cdots,x_k$. ...
4
votes
0answers
83 views

Symmetric group isomorphism

Let $G$ be a group generated by $a$ and $b$ such that order$(a)=2$ and order$(b)=n$. What condition can I impose to $a$ and $b$ to get $G \cong S_n$? The group $G$ that I'm looking at is a little ...
3
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5answers
1k views

Definition of presentation of a group.

basing myself on suggestions I found in previous discussions, I have opted for the algebra book of Dummit. However, I have now a little problem. On page 26 (3th edition) the authors say that "...
3
votes
1answer
77 views

Quaternion Group: Determine that $i^4 = 1$.

Suppose we are given the following presentation of the quaternion group: $Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$ Is it obvious that $i^4 = 1$?
3
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3answers
651 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 $\mathbb Z$...
3
votes
3answers
2k views

Order of a Group from its Presentation

Let $G$ be a group with generators and relations. I know that in general it is difficult to determine what a group is from its generators and relations. I am interested in learning about techniques ...
3
votes
1answer
777 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of "...
3
votes
3answers
223 views

How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
3
votes
2answers
118 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 ...
3
votes
2answers
98 views

Find the group given the presentation

I have the group presentation $$ G := \langle a,b |a^8=b^8=1,a^{-1}ba=b^{-1},b^{-1}ab=a^{-1} \rangle. $$ Which group is it? Notably, what about its order?
3
votes
2answers
236 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}...
3
votes
1answer
180 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 ...
3
votes
1answer
341 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?
3
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2answers
183 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 type ...
3
votes
3answers
60 views

How to prove that this group has at most order 16

If $$G=\langle a,b:a^8=b^2a^4=ab^{-1}ab=e\rangle,$$ how can I prove that $G$ has order at most $16$? I have played with the relations for a while, but am literally stuck. I know that the order of ...
3
votes
3answers
162 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?...
3
votes
1answer
1k views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by $$D_n=\{I,...
3
votes
1answer
30 views

Focal subgroup is normal

When $H$ is a subgroup of $G$, we can define the focal subgroup of $H$ as $$H^\ast:=\langle h^{-1}h'\mid h,h'\in H, h'=h^g, g\in G\rangle.$$ I'm confused about the proposition "$H^\ast$ is a normal ...
3
votes
1answer
54 views

Is it generally true that $\langle a,b \rangle \cong \langle c,d \rangle\Rightarrow \text{ either }|a|=|c|,|b|=|d| \text{ or } |a|=|d|,|b|=|c|$?

Background: We are given two groups $G,H$ generated by two elements, say $G=\langle a,b\rangle$ and $H=\langle c,d\rangle$. Further suppose that the orders of $a,b,c,d$ are finite and $\{|a|,|b|\}\neq\...
3
votes
1answer
97 views

What is the name of this normal subgroup of the free group?

Given the free group $F_S$, we can define a normal subgroup of $F_S$ as follows. For a given element $w \in F_S$ and $s \in S$, define "the count of $s$", $c_s(w)$, as the number of times $s$ appears ...
3
votes
1answer
267 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
3
votes
2answers
109 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 ...