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Questions tagged [group-presentation]

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

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Is every finite group an arithmetic group?

Lubotzky-Phillips-Sarnak(LPS88) constructed Ramanujan graphs of degree $d=p+1$ (for an odd prime $p$) as Cayley graphs of projective linear groups with respect to a carefully chosen set of generators ...
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103 views

Why is $H(r, 4, r-1)\cong \Bbb Z_5$ for $r=2, 3\pmod{4}$?

I'm due to start my PhD this October and I'll be working closely with $H(r, n, s)$, so a detailed answer aimed at that level would be great. The Details. Definition 1: Let $w=x_1\dots x_r(x_{r+1}\...
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What is a rewriting process for a subgroup?

I start my PhD in Mathematics this October and I'll be working closely with presentations, so a detailed answer aimed at that level would be ideal. I want to understand the beginning of Section 2.3 ...
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204 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-...
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1answer
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How is $I(r, n, s)$ a semi-direct product of $H(r, n, s)$ with $C_n$?

I'm due to start my (fully funded!) PhD in Mathematics this October (2017) and I'll be working closely with $H(r, n, s)$, so a detailed answer aimed at that level would be ideal. The Details: ...
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53 views

Family of groups with specific presentation

Is there a name for the family of groups given by $n$ generators ($g_1, g_2,\ldots g_n$) and the following relations? $$g_ig_jg_i = g_jg_ig_j,~\forall i,j \in \lbrace 1,\ldots n\rbrace,~i\neq j\\ ...
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Let $G=\langle X, Y|X^p=Y^q=(XY)^r=1, XY=YX\rangle,$ for $p$ prime, $p\leq q\leq r$. Show that if $p\nmid r$ then $G=C_b$ for $b=gcd(q,pr)$.

I have a group $G$ that has presentation $$\langle X, Y \mid X^p=Y^q=(XY)^r=1, XY=YX \rangle,$$ where $p$ is prime such that $p\leq q \leq r$. I need to show that if $p \nmid r$ then $G = C_b$ where $...
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1answer
173 views

The normal closure is the smallest normal subgroup such that every element of $R$ is identified with the identity

The normal closure $N_R$ of a subset $R$ in a group $G$ is the subgroup generated by $\bar R=\{g^{-1}rg|g\in G,r\in R\}$. I read it is the smallest normal subgroup such that every element of $R$ is ...
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106 views

Prove the Higman group is infinite but a similar one is trivial

(4) shows the difficulty of proving a finite presentation group is trivial or not. Moishe Cohen mensioned the first group is called Higman group. I can't see why the Higman group is infinite but the ...
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1answer
85 views

The commutator subgroup in terms of presentations.

This is Exercise 2.1.8 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Details: Theorem 1: Let $G=\langle a, b, c, \ldots \...
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1answer
79 views

The normal subgroup of $\langle a, b\mid a^{22}, b^{15}, ab=ba^3\rangle$ generated by $a^2$.

This is Exercise 2.1.5(b) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Question: For each of the following groups $G$, let $H$...
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2answers
85 views

The normal subgroup of $\langle a, b\mid a^4, a^2=b^2=(ab)^2\rangle$ generated by $a^2$.

This is Exercise 2.1.5(a) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Question: For each of the following groups $G$, let $H$...
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1answer
56 views

The index of the normal subgroup generated by $a^3, b^2, aba^{-1}b^{-1}$ in $\langle a, b\rangle$.

This is Exercise 2.1.4(d) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
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The normal subgroup generated by $a^2, b^2$.

This is Exercise 2.1.4(c) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
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1answer
52 views

Presentation of $Gal\left( \mathbb Q(\sqrt[8]{2},i)/\mathbb Q\right)$

Let $E = \mathbb Q(\sqrt[8]{2}, i)$ and $\zeta$ be a primitive root of unity. We have $[E:\mathbb Q] = 16$. The possible elements of $G := Gal\left(E/\mathbb Q\right)$ are: $$\begin{cases} \sqrt[8]{2}...
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1answer
67 views

A normal subgroup of $\langle a, b\rangle$.

This is Exercise 2.1.4(b) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
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1answer
55 views

Finding the index of $N$ in $F$.

This is Exercise 2.1.4(a) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
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355 views

Proof that dihedral group $D_{2n}$ is isomorphic to the group generated by two group elements of order 2

An exercise in Aluffi asks the reader to prove that if $a,b$ are distinct elements of order 2 in a group $G$ and $ab$ has finite order $n\geq3$, then the subgroup of $G$ generated by $a,b$ is ...
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1answer
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In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$?

The Question: In $\langle a, b\mid a^2, b^3, (ab)^2\rangle$, why does $ba=ab^2$? My Attempt: Clearly the presentation defines the group $\mathcal S_3$ under the isomorphism given by $\theta: a\...
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1answer
118 views

Tietze Transformations: $\langle a, b, c\mid b^2, (bc)^2\rangle$ and $\langle x, y, z\mid y^2, z^2\rangle$.

This is Exercise 1.5.2 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Details: A Tietze transformation of one presentation $$\...
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Presentation of $SL_2(p^m)$, for $p$ odd prime

For $n \ge 2$, $p$ prime greater than 2, and $m \ge 1$, let $SL_n(p^m)$ be the Special Linear group of degree $n$ over the finite field of order $p^m$ (that is, $n\times n$ matrices with entries in $\...
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1answer
158 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 ...
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2answers
94 views

Showing a presentation is a permutation group.

This is Exercise 1.2.8 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Question: Show that the group $$\langle a, b, c\mid a^3,...
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280 views

Group Presentation of the Direct Product.

This is Exercise 1.2.5 and Exercise 1.2.6 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Details: Definition 1: Let $$\langle a,...
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1answer
86 views

Showing a group is Abelian using its presentation.

This is Exercise 1.2.2 of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Show that $$G=\langle a, b, c\mid P, Q, R, \...
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Find a nontrivial presentation of this group of permutations: the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$.

The Question: Find a nontrivial presentation of the permutation group $G$ consisting of the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$. My Attempt: Under the ...
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60 views

A presentation of $C_4$ with one generating symbol $b$ and two defining relators.

This is Exercise 1.1.5(c) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al. The Question: Give a presentation of $C_4$ using one ...
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Cyclically Presented Groups.

I suppose I've answered my own question in writing it but some input would be great. The Question(s) What exactly are cyclically presented groups, what are some examples, and where might I find ...
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Check if an especific equation is a consequence of others (noncommutative scenario)

Let $\Sigma = \{a, b, c, ..., x, y, w\}$ be a set of formal variables, $\Sigma^{-1} = \{a^{-1}, b^{-1}, ..., w^{-1}\}$ be the inverses of those variables, and $B = \{ e_1, e_2, ..., e_n \}$ be a set ...
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1answer
163 views

Outer automorphism of A4 via automorphism of a free group

The question is about an exercise I found in Robinson's book A Course in the Theory of Groups. It says that, given a presentation $\pi$ from a free group $F$ to a group $G$ and said $R=$ker$\pi$, one ...
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1answer
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Abstract algebra: finding order or group [closed]

Find the order of a group defined by: $$G=\{u,v\,|\,u^4=v^3=1,\,u\cdot v=u^2\cdot v^2\}.$$ When I first looked at the problem it looked like dihedral group but it's not. I am unable to find ...
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Notation and definition on book Presentations of Groups

On the 33th page of the book Presentations of Groups, when talking about the Nielsen's Method to prove Nielsen–Schreier theorem, the authors consider $F$ as any free group, $H$ as any subgroup of $F$, ...
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48 views

Presentation of (presumably) inner direct product $G_1\times G_2$

Let $G_1=\langle S_1\mid R_1\rangle,\ G_2=\langle S_2 \mid R_2 \rangle $ and $S_1\cap S_2=\emptyset$. Show that $G_1\times G_2=\langle S_1\cup S_2\mid R_1\cup R_2\cup R\rangle$ where $R=\{S^{-1}T^{-1}...
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1answer
117 views

Understanding statement: F free of rank r $\Rightarrow$ F/F' free abelian of rank r

I found the following exercise on the 12th page of Presentations of Groups: Let $F$ be free of rank $r$. Show that $F/F'$ is free abelian of rank $r$. First of all, $F'$ is not defined. I'm ...
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Does $X_{2n}=\langle x,y|x^n=y^2=1,xy=yx^2\rangle$ really have order $6$?

Here is the entire question: Let $X_{2n}=\langle x,y\mid x^n=y^2=1,xy=yx^2\rangle $ and $n=3k$ then show that $X_{2n}$ has order $6$. Here is my question: How do we know that $X_{2n}$ has ...
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How to check that the Braid group has the known presentation?

Why we are sure that the presentation for the braid group is $$ \mathcal{B}_n= \left\langle \sigma_1 , \dots, \sigma_{n-1} \vert \mathcal{R}_n \right\rangle, $$ where $$\mathcal{R}_n=\left\{ \begin{...
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Group presentation of a direct product.

Suppose $G_1=\langle X_1|R_1\rangle$, $G_2=\langle X_2|R_2\rangle$, $X_1\cap X_2=\emptyset$. I want to show that $G_1\times G_2=\langle X_1\cup X_2|R_1\cup R_2\cup[X_1,X_2] \rangle$. By using ...
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1answer
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$G=\langle S|R\rangle$ is finitely presented simple, then if $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$

I'm having some trouble seeing $G=\langle S|R\rangle$ is finitely presented and simple, then if the world in free group $F(S)$, $w\neq e$ in $G$, $\langle\langle w\cup R \rangle\rangle = F(S)$ ...
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1answer
44 views

Reducing $ \langle a,b \vert ab^{-1}ab^{-1}=1 \rangle $ into more useful form.

I am working on a question and have found the group presentation in the form of $$ \langle a,b \vert ab^{-1}ab^{-1}=1 \rangle $$ I believe I need to reduce this further but am unsure how. I've tried ...
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Presentation of the shift-replace monoid

Let \begin{align} S(f)(x) &= f(x + 1) \\ R_a(f)(x) &= \begin{cases} a & x = 0 \\ f(x) & \text{otherwise} \end{cases} \end{align} What is the ...
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Showing $\Bbb{Z}_q \rtimes Q_8$ has the presentation $\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$

Let $G \cong \Bbb{Z}_q \rtimes Q_8$. Then $G$ has a presentation as follows $$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$$ I dont understand why $x^z=x^{-1}$? ...
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1answer
252 views

Prove the existence and uniqueness of a group homomorphism

Let $G_1$, $G_2$ be arbitrary groups, and $H$ be any group with homomorphisms $\theta_1:G_1\rightarrow H$, $\theta_2:G_1\rightarrow H$. Show that there exist a group $G$ and homomorphisms $\beta_1:G_1\...
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1answer
269 views

Group presentation for discrete Heisenberg group

I have to show that the discrete Heisenberg group H (with $a,b,c \in \mathbb{Z}$) has the presentation $H=\langle x,y\mid [[x,y],x]=[[x,y],y]=1 \rangle$. I figured we write $H$ as a group of triples $...
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1answer
142 views

How to prove $\mathbb{Z}_4 \times \mathbb{Z}_2$ is isomorphic to $\langle x, y \mid x^4=y^2=1, x^y=x \rangle$?

Here is my attempt at proving $$\mathbb{Z}_4 \times \mathbb{Z}_2= \langle x, y \mid x^4=y^2=1, x^y=x \rangle$$ Let $F$ be a free group upon $\{x,y\}$. Define a homomorphism $\theta: F \rightarrow \...
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64 views

Given a presentation of a group, find its normalizer.

Let $F$ and $H$ be two free groups, where $F = \langle a,b\rangle$ and $H = \langle aa,bb,aba,baab, babab\rangle$. Let $N(H)$ be the normalizer of $H$ in $F$. How can I compute $N(H)/H$? In general,...
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1answer
522 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,...
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2answers
150 views

How to prove that $\langle x, y \mid xyx^{-1}y^{-1}\rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$

I am a bit new to presentations and free groups, so of course I am stuck on a very easy question. I want to prove that $\langle x, y \mid xyx^{-1}y^{-1}\rangle \cong \mathbb{Z} \times \mathbb{Z}$. ...
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1answer
111 views

Extending a map to a well defined homomorphism

Given a map $f: S \rightarrow H$, with $S$ some set of generators, when and how can $f$ be extended to a well defined homomorphism with $\phi : G \rightarrow H $ with $G$ a group generated by $S$ with ...
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1answer
199 views

Presentation of wreath product $G=S_3 \wr S_3$ of symmetric groups. What is the isomorphism type of $G/[G,G]$?

I'm trying to answer the first part of a group theory question as revision for an exam that goes as follows; Let $G = S_3 \wr S_3$, the permutational wreath product of two symmetric groups of ...
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1answer
172 views

Definition of Lie algebra presentations

A group presentation is defined as a collection of generators (such that every element is a product of powers of those) and relations between the generators such that the group is uniquely defined by ...