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Questions tagged [group-presentation]

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

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Meaning of a group homomorphism to a symmetric group

I trying to understand the meaning of a group homomorphism to a symmetric group. Let's say we have a finitely presented group $G:=\langle X \mid R \rangle$ with $X=\{x_1, \dots, x_n\}$ and the ...
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Which of these numbers could be the exact number of elements of order $21$ in a group?

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.46. Which of the following numbers could be the exact number of elements of order $21$ in a group: $21600, 21602, 21604$?...
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Quotient group of the dihedral group by $\langle r^2 \rangle.$

Show that $G/H$ is abelian, where $G$ is the dihedral group $$ G={\langle r,\, f \mid r^n=f^2=1,\, rf=fr^{-1}\rangle}$$ and $H$ is the subgroup $\langle r^2 \rangle.$ I've tried showing that for $...
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Finite Presentation of a subgroup

I have the group $\langle a,b \mid a^3b^3\rangle$ Now I send both $a$ and $b$ to the generator of $\mathbb{Z}/3\mathbb{Z}$. This gives a well-defined homomorphism from our group to $\mathbb{Z}/3\...
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Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?

I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.) Let $G$ be a group, and let $a,b,c\...
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Understanding semidirect product by constructing a non-abelian group of order $21$

I just learnt semidirect product, but only know the basic definition, not gaining the true understanding of it. There is an example that asks the reader to construct a nonabelian group of order $21$. ...
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Show that $\langle a , b | a^2 = b^3 = e \rangle$ is infinite and nonabelian. [duplicate]

Prove that the group $G$ defined by generators $a,b$ and relations $a^2 = b^3 = e$ is infinite and nonabelian. This is a question from Algebra by Hungerford, Section I.9, page 69. This question may ...
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1answer
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Presentation of $A_5 \times \Bbb Z_2$.

I want presentation of the group $A_5\times \Bbb Z_2$ which is a group of order $120.$ I know the presentation of $A_5$ but not of product. I tried it in GAP . In GAP its Atlas name is $2\times A_5$ ...
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Why do generating sets need not contain the inverses of their elements?

I recently learned about generating sets, and a common elementary example that is provided is the sets $\{1\}$ and $\{-1\}$, both of which independently generate $(\mathbb{Z},+)$. I understand why ...
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Deriving the Discrete Heisenberg Group generators.

How can we derive the generators of the Discrete Heisnberg Group? Everyone seems to just state this as a given and never actually derive it from scratch. I'm looking for a (somewhat) elementary ...
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I am looking for a modern and thorough exposition for presentations of groups

Conider the following abstract description for the Quaternion group: $$\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\rangle$$ This description is called a presentation of the Quaternion group ...
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Showing affine transformations group generated by $2x$ and $x+1$ is the Baumslag-Solitar group.

I want to compute the presentation groups of $\langle f,g\rangle$ the generated group of affine transformations with $f(x)=2x$ and $g(x)=x+1.$ The affirmation is $\langle f,g\rangle=\langle a,b\...
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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. $...
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Show that $\mathbb{Z} = \langle a, b \mid a^{12} = b,\ ab = ba \rangle$ has dead end elements

This exercise is taken from the book "Office Hours with a Geometric Group Theorist" (Office Hour 15, exercise 8): Exercise: Show that the group $\mathbb{Z}$ has dead end elements with respect to the ...
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Presentations of $D_8$ using permutations

I fond a lot of examples using presentation of $D_8$ by generators which are permutations of $S_4$. 1) How many presentations could be found? 2) Could it be presented by permutations of $S_5$ or ...
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Let $G$ be a group and suppose that $a,b\in G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove that $b^5=1$. [closed]

I recently found an exercise about group presentation that I have no idea how to work out. Can anyone help me? Let $G$ be a group and suppose that $a,b\in G$ satisfy $a^2=1$ and $ab^2a=b^3$. Prove ...
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Calculating a presentation of $\mathbb{Z}_{3}$ in detail.

Theorem: Let $G$ groups and $S\subset G$ such that $\langle S\rangle =G$. (Here $G=\left\{s_1\ldots, s_n:s_i\in S\cup S^{-1}, n\in\mathbb{N}\right\}$.) Let $\varphi:S\to G$ with $\varphi(s)=s$. By ...
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What is a presentation of the upper triangular subgroup of $GL(2, \mathbb Q)$?

I have been trying to find a presentation of the upper triangular subgroup of $GL(2, \mathbb Q)$ by considering the free group $Fr(\{x_i| i\in \mathbb Q\})$ under a homomorphism $f$ into $GL(2, \...
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Cayley graph of Rubik's cube group

(a) I would like to know whether there is a group theoretic approach for calculating the diameter of the Cayley graph of Rubik's Cube group. I know it's been proved that the above diameter is $20$ ...
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Presentation of the quaternion group $Q_{16}$

I was asked to prove that $\langle x,y|x^8 = 1 , x^4 = y^2 , xy = y^{-1} x\rangle$ defines a $2$-group of order at most $16$. It is well-known that the group $\langle x,y|x^8 = 1 , x^4 = y^2 , xy = y^{...
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Automorphism group of $\mathbb Z_2^3$

I am trying to find $\text{Aut}(\mathbb Z_2^3)$ and express it in terms of familiar groups and the direct and/or semi direct product. Here's what I have so far: I know that the set of generators $A:=\...
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Finitely presented groups which are neither Hopfian nor cohopfian

Are there any examples of (preferably countable) finitely presented groups which are neither hopfian nor cohopfian? If so, is there a classification of such groups?
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Finding the order of a presentation of a group

Show that $M = \langle r,s \mid r^m = e, s^n = e, srs^{-1} = r^j \rangle$, where $j$ is a natural number satisfying $\operatorname{gcd}(j,m) = 1$ and $j^n \equiv 1 \pmod{m}$, has $mn$ elements. I'm ...
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Minimal size of a generating set for presentations of finite groups

Are there any results on the minimal number of generators required to give a presentation of a finite group? More specifically, given a group G, what is the minimal number of generators needed for a ...
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Showing formally that $H:=\langle x,y| x^2, y^n, yxyx^{-1} \rangle$ is a presentation of $D_{2n}$

I want to Show formally that $H:=\langle x,y| x^2, y^n, yxyx^{-1} \rangle$ is a presentation of $D_{2n}$. To start with, by the universal property of the free group, there is a group homomorphism $\...
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Two presentations of a group, one certainly finite. Need the other be?

I know the answer to the question above is "no", quite flatly. The counter example is below: $$\mathbb{Z}\cong\langle a,b\mid b^2a^{-1}\rangle\cong \langle a,b\mid\lbrace b^{2^{n+1 }}a^{-2^n}:n\in\...
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$G=\langle a,b \mid baba^{-1}=1\rangle$ Show that $\langle a \rangle$ is infinite

Let $G=\langle a,b \mid baba^{-1}=1\rangle$. Show that the subgroup generated by $a$ is infinite. My attempt Suppose $\langle a\rangle$ is finite so $a^k = 1$ for some $k \in \mathbb{Z}$. So I ...
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Sharpness of the constant $1/6$ in Cancellation Theorem.

Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $R = R^{-1}$ of freely and cyclically reduced relators, and let $\Lambda$ be the girth of $\langle \; S \; | \; R \; ...
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Trefoil Knot Group

I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling ...
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Embedding of torsion free nilpotent group into Unitriangular matrix group $UT_n(\mathbb{Z})$

I have given the presentation $ G = \langle a_1,a_2\ |\ [[a_i,a_j],a_k],\forall i,j,k\in{1,2},\ a_1^2 = a_2^2 \rangle $ of a torsionfree, nilpotent group which looks pretty similar to the Heisenberg ...
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How to prove that two groups with different presentations are isomorphic in a naive way?

One can define a presentation of a group naively (ala Dummit-Foote in Chapter 1.2), i.e., as a group generated by certain elements with certain relations such that all other relations follow from the ...
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Inequivalent Canonical words

I have a problem in understanding one line of "Combinatorial Group Theory" by Magnus, Karrass and Solitar (Page 27-28). (Please find it in the image below) In equation (4), it considers a group $G=\...
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What is the meaning of presentation of an unital associative Ring?

Let $R$ be an unital associative ring and let $f: F \rightarrow R$ be an onto ring homomorphism. Where F is some freely generated ring over the set $S$ then $<S|T>$ is called presentation of ...
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Show that the group $G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$ has order $27$.

This is Exercise 1.2.21 of Magnus et al's book on combinatorial group theory. The Question: Show that the group $$G=\langle a, b\mid a^3, b^3, c=b^{-1}a^{-1}ba, ac=ca, bc=cb\rangle$$ has order $27$...
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How to reconcile two different computed first homology groups for this space?

I am asked to classify the compact surface obtained by pasting the edges of a polygonal region with the labeling scheme $abcdabdc$ and compute it's first homology group. I classify the space as the ...
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Groups of order $2p$

There are a few question on classification of groups of order $2p$ on MSE but I'd like to receive a feedback on this proof (and have a question about it at the end). Let $G$ be a group of order $2p$. ...
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Reference request fundamental group of surface of genus $g$ and $n$ boundary components

Let $\Sigma_{g,n}$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $\partial_i$. I would like to find a reference for the ...
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Burnside groups : a few questions

Let $G=B(n,e)=F_n/\langle\langle F_n^e\rangle\rangle$ be the Burnside group on $n$ generators with exponent $e$, i.e. the quotient of the free group on $n$ generators $F_n$ by the normal subgroup ...
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Is there a foolproof method to calculate the inverse of a group homomorphism?

For example, take the group $\langle a,b\mid a^{-1}b^2ab^{-3}\rangle$, and the homomorphism given induced by the map $a \rightarrow a$, $b \rightarrow b^2$. Is there a method that will let you ...
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Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$

When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following: $\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$ See https://groupprops....
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Identifying $\langle a,b\mid a^2b^2\rangle$.

The Question: What group is $G=\langle a,b\mid a^2b^2\rangle$? Thoughts: I found that the presentation maps onto $\langle a, b\mid a^2, b^2, 1\cdot 1\rangle\cong \mathbb{Z}_2\ast\mathbb{Z}_2$, ...
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Describing $\langle x,y : x^{2} , y^{3} , [x,y] , x^{6}y^{6}\rangle$.

I am trying to identify the following presentation $\langle x,y : x^{2} , y^{3} , [x,y] , x^{6}y^{6}\rangle$ I substituted the first relation in the final one and got $x^{6}=1$ so the group is ...
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Identifying group structure of matrix group

I am working on classifying the stabilisers of quadratic binomials in $GL(\mathbb{C}^{n})$ but struggling to identify the groups which are appearing. One example is the binomial $$x_{1}x_{2}-x_{1}x_{...
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Efficiently calculating the conjugacy classes of a finite group with a certain type of presentation.

I am regularly attempting to determine a list of conjugacy classes of a finite group $G$, with a presentation that looks something like $$\langle a,b \mid a^r = b^s = 1, b^{-1}ab=a^t \rangle, $$ ...
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How to add relations to make groups trivial

Suppose I have a group $G = <x, y|x^2y^5>$, and I want to add a single relation to make G trivial. Is there any way to do this? In general, if we have a group $G = <x, y|r>$, where r is a ...
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1answer
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Central extension of an Abelian group

How to prove that in the group described by the presentation $$\langle a,b\,|\,a^{p^3}=b^{p^3}=[[a,b],a]=[[a,b],b]=1\rangle$$ the commutator $[a,b]$ has order $p^3$? I cannot find a reasonable method ...
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1answer
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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?
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Defining Groups via the Universal Property

The snippets below on group presentation led to the following questions: What is the explanation/intuition of the universal property. Basically what (2) is saying. I understand free groups (though ...
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1answer
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Uniqueness of group given only group presentation

I'm trying to understand how a given presentation of a group is well defined. It says on Wikipedia (https://en.wikipedia.org/wiki/Presentation_of_a_group) that $G \ =\ \langle a\ |\ a^n=1\rangle$ is a ...
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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 ...