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Questions tagged [group-presentation]

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

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Presentation of a group generated by reflections through hyperplane

Question: Let $P_i\in\mathbb{R}^n$ be the hyperplane $x_i - x_{i+1} = 0$. Find a presentation for the group $G$ generated by the reflections in $P_1, \ldots, P_{n-1}$ Attempt: I really don't know how ...
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Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (...
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Find the isomorphism type

Consider the abelian group $G$ generated by $a$, $b$ and $c$ and determined by the following relations \begin{aligned} 3 a+9 b+9 c &=0 \\-3 b+9 c &=0 \end{aligned} determine the isomorphism ...
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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\...
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Is there a way to describe the structure of $Aut(UT(3, p))$?

Is there a way to describe the structure of the automorphism group of $$C_{p^2} \rtimes C_p \cong \langle x, y, z | [x,y]=z, [x,z]=[y,z]=x^p=y^p=z^p=e \rangle \cong UT(3, p)?$$ Here $p$ is an odd ...
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Is this proof that $\widehat{G_p}$ is pro-$p$ free correct?

Let $G$ be an abstract group with the following presentation: $$G \simeq \langle x,y \mid x^2y^2 = 1 \rangle $$ Let $p \neq 2$ be an odd prime. I want to show that $\widehat{G_p} \simeq \mathbb{Z}_p$...
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Symmetrising the relations in a presentation of a group

Let $G$ be a finitely presented groups defined by $$G=\{x_1,\ldots,x_n\mid R_1(x_1,\ldots,x_n)=\cdots=R_m(x_1,\ldots,x_n)=1 \}.$$ Let this presentation be denoted by $P$. Let $S_n$ be the ...
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1answer
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How to show that $|D_{2n}| = 2n$ via the presentation?

Consider the dihedral group $$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$ How can I show that $|D_{2n}| = 2n$? I'm trying to show that we can write every element in the form ...
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$p$-completion is pro-$p$ free

Let $G$ be an abstract finitely generated residually finite group, and suppose that it's $p$-completion $\widehat{G_p}$ is a pro-$p$ free group. Does this implies that $G$ is a free group? The ...
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If we are handed the presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group?

If we are handed the group presentation $\langle i,j,k \mid i^2=j^2=k^2=ijk \rangle$ and nothing more, can we deduce that this is the quaternion group? Nothing in this presentation tells us that $i^2=...
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Proving the Free Abelian Group is Free Abelian…?

On page 40 of these notes is the following exercise: Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$. On ...
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Can I conclude that my group is finitely generated, if it is a homomorphic image of a free-group on finitely many generators?

Say $X$ is a finite set, $F \langle X \rangle$ is the free group on the set $X$ and $G$ be a group. If I have a surjective homomorphism $$\varphi : F \langle X \rangle\longrightarrow G$$ then can I ...
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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 $i^4 = 1$?
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What is an algorithm for determining if a finitely presented group is finite

Suppose I am given a presentation of a group with a finite number of generators and a finite number of relations. Is there an algorithm for determining if the group is finite? Also, if there is such ...
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Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?
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How do I turn a group presentation into a multiplication table?

Suppose I have a group presentation and I know for a fact that this is a presentation of some finite group. How do I create a multiplication table from this presentation?
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1answer
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A proof that $\langle u,v\mid u^4=v^3=1, uv=v^2u^2\rangle$ defines the trivial group.

This appears to be new to MSE. I'm reading "Abstract Algebra (Third Edition)," by Dummit & Foote. This is based on Exercise 1.2.18. Question: Show that $$Y=\langle u,v\mid u^4=v^3=1, uv=v^2u^...
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Derived subgroup of $\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$. [closed]

I would like to prove that if $M_n(p)=\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$, then $M'_n(p)$ is a cyclic group of order $p$. I was wondering if someone could ...
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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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2answers
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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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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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3answers
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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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1answer
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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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1answer
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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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1answer
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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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3answers
100 views

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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1answer
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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 ...