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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group presentation with circularly shifted relator

Let $G=\langle S \mid R_1 \cup R_2 \cup R_3 \rangle$ be a group presentation with $S=\{a,b,c\}$, $R_1=\{aa{^{\text{-}1}}, bb{^{\text{-}1}}, c^2\}$, $R_2$ the set of all circular shifts of the word $w=...
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Is there an algorithm that can compute finite presentations for finitely presentable subgroups in a FP group with solvable word problem?

Given a group and its finite presentation $G=\langle A\mid R\rangle$, I want the following algorithm: Input: a finite set $W$ of words in $A\cup A^{-1}$ that generates a finitely presentable subgroup ...
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1 vote
2 answers
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Realization of the Modulor group $M_{16}$ of order $16$. [closed]

Consider the Modulor group $M_{16}$ of order $16$ given by the presentation $$ M_{16}=\langle a,b:a^8=b^2=1,ba=ab^5\rangle $$ Is there a known realization of $M_{16}$ by matrices?
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Defining semidirect product and presentation when one of the groups is a product of cyclic groups

I'm trying to classify the groups of a certain order and have shown that a group $G$ with that order can be expressed as the semidirect product of a normal subgroup $N$ $\cong$ $C_n$ $\times$ $C_m$ ...
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Exponential notation for Galois group and egality

Let $G$ a non abelian group of order 20 generated by two elements $\sigma$ and $\tau$ such that $G = \langle \sigma, \tau |\,\, \sigma^5 = \tau^4 = 1,\, \tau\sigma = \sigma^3\tau,\, \sigma\tau = \tau\...
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Dummit & Foote Chapter 1.2 Exercise 17 (Group Presentations)

The exercise is to show that if $n = 3k$, then the group presentation $\langle x, y \mid x^n = y^2 = 1,\ xy = yx^2 \rangle$ generates $D_6$. The group presentation for $D_6$ given in the book is $D_6 =...
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Reference request: proof of the Rips Construction

I'm trying to understand how the Rips Construction works. In particular, I'd like to understand why the presentation cooked up by the Rips construction (which if I'm not mistaken is not explicitly ...
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3 votes
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For any finite group $H$ and homomorphism $\alpha:BS(2,3)\to H$, prove $\alpha([bab^{-1},a])=1$

$BS(2,3)=\langle a,b \mid ba^2b^{-1}=a^3 \rangle$. Let $H$ be a finite group and $\alpha:BS(2,3)\to H$ a homomorphism. Let $g=[bab^{-1},a]$. Prove $\alpha(g)=1$ My Attempt $$\begin{align} \alpha(g)&...
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Finding relators of a matrix group

Let $f_1,\dots,f_n$ be maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $f_i(x) := a_ix + b_i$ with $a_i,b_i \in \mathbb{Q}$. We construct the transformation group $G = \langle f_1, \dots, f_n \...
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Is this a useful (and, more importantly, correct) way to think about $\operatorname{Dic}_3$?

I know that there's already a question here asking about the geometric interpretation of a dicyclic group, and at risk of running headfirst into the Dunning-Kruger effect, I think I came up with ...
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If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, can the following abelianization be finite?

Suppose that we have a finitely generated group $G = K\rtimes\mathbb{Z}$, but $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0)\mid k \in T \rangle$ is a ...
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Why must a group with presentation $\langle r,s \mid r^n=s^2=1,rs=sr^{-1}\rangle$ have exactly $2n$ elements?

In Dummit and Foote's Abstract Algebra, they state on page 27 of chapter 1 that any group with the presentation $\langle r,s \mid r^n=s^2=1,rs=sr^{-1}\rangle$ must have exactly $2n$ elements. Why is ...
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How to find the discrete Heisenberg group in this group given by a small presentation?

Consider the group $G$ given by the following presentation: $$G=\langle x,y\mid x^{-1}y^2xy^2=x^{-2}yx^{-2}y^3=1\rangle.$$ In this slides it is noted that this is a torsion-free polycyclic group, ...
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What is the group presentation for the two-twist spun trefoil?

The sources I'm looking at are giving me conflicting information. One paper gives the presentation $$\langle x,y|xyx=yxy, x^2y=yx^2\rangle,$$ while another paper asserts that Example 12 from Fox's A ...
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Another presentation of a surface group

Let $S$ be a connected orientable closed genus $2$ surface. Its fundamental group admit a presentation which is $$\pi_1(S)=\langle a,b,c,d\ |\ [a,b][c,d]\rangle.$$ This presentation is related to the ...
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Defining relation on semidirect product of groups

$\newcommand{\aut}{\operatorname{Aut}}$ I wonder how we can extract defining relation on semidirect product of groups. Consider a group of order $12$. Consider two cyclic groups $C_3 = \langle a\...
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Isomorphism between group presentations [closed]

Let $$G_1=\langle a,b,c:\hspace{0.2cm}abc^{-1}a^{-1}b^{-1}c=1\rangle,$$ $$G_2=\langle d,e:\hspace{0.2cm}de d^{-1}e^{-1}=1\rangle.$$ How can I construct (if possible) a group isomorphism between $G_1$ ...
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A space $X$ that has fundamental group with a presentation [duplicate]

Consider the group $G$ with presentation $$G=\langle a,b:\hspace{0.1cm}a^2b^{-1}a^3b^2=1\rangle.$$ What is the explicit path-connected topological space $X$ with $\pi_1(X)\cong G$? My attempt: I try ...
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On the Frobenius group

I am studying some properties of Frobenius group $G$ of order $20$, which mean that a presentation of the group $G$ is $$G = \langle c, f \mid c^5 = f^4 = 1, \,cf = fc^2\rangle.$$ My question is ...
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Simplicial homology group in degree $1$ of a certain $\Delta$-complex

Suppose that we have the $\Delta$-complex $X$ (shown below), with $0$-simplices $p$ and $q$, $1$-simplices $a$,$b$,$c$ and $d$, and $2$-simplices $U$,$V$ and $W$. I'd like to compute $H_1^{\Delta}(X)$...
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Counting the size of a supposed presentation of $S_4$

I would like to show $$P :=\langle a,b \, | \, a^2,b^4,(ab)^3\rangle \cong S_4.$$ I'll assume $$S_4 \cong G:= \langle x,y,z \, | \, x^2,y^2,z^2, (xy)^3, (yz)^3, (xz)^2 \rangle.$$ Let $\phi: G \to P$ ...
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Given $H=\langle X_H\mid R_H\rangle$ and $K=\langle X_K\mid R_K\rangle$, find a presentation for Robinson's $H\circ K$

According to this search, this question is new to MSE. The Details: Paraphrasing Robinson's, "A Course in the Theory of Groups (Second Edition)", we have the following Definition: Let $G$ ...
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Show ${\rm Aut}(G)$ is a $2$-group, where $G$ is given by a particular presentation

This is Exercise 5.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The exercise is marked as being referred to later on in ...
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Find the upper central series of $Q_{2^n}$.

This is part of Exercise 5.3.3 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The Details: (This can be skipped.) On page 125 of ...
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Presentation of the trivial group

Dumb question: Is $\langle e \mid e\rangle$ a presentation of the trivial group?
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Let $G = \langle x, y \mid x^{7} = y^{3} = e,\; yxy^{−1} = x\rangle$. What is $|G|\,$?

Let $G = \langle x, y \mid x^{7} = y^{3} = e,\; yxy^{−1} = x\rangle$. What is $|G|\,$? What I've done so far: $$yxy^{-1} = x \implies yx = xy.$$ The group $G$ is an abelian group. I can only think ...
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Order of group given by the presentation $\langle \{a\}\mid \{a^r,a^s\}\rangle $ where $r$ and $s$ are coprime

I want to find the order of a group given by the presentation $\langle \{a\} \mid \{a^r,a^s\}\rangle $ where $r$ and $s$ are coprime. My reasoning is as follows: since $a^r=1$ and $a^s=1$, the order ...
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Does this group construction preserve finite presentability?

Suppose $G$ is a group. Consider the set $G^G$ of all functions $G \to G$, which forms a group under elementwise multiplication. Now, for all $g \in G$ let’s define $c_g \in G^G$ as the constant ...
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Let $G=\langle x, y\mid x^7=y^3= e, yxy^{−1}=e\rangle$. Find $|G|$. Find a group that is isomorphic to $G$. Explicitly state the isomorphism

Let $G = \langle x, y \mid x^{7} = y^{3} = e, yxy^{−1} = e\rangle$. What is $|G|$? Find a familiar group that is isomorphic to $G$. Explicitly state the isomorphism. I am trying to get a handle on ...
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8 votes
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Generalisation of the Symmetric Group

For $m\in\mathbb{N}$, consider the group $G_m=\langle s_1,\dots,s_{n-1}\rangle$ generated by the relations \begin{align*} s_i^m&=1\\ s_is_j&=s_js_i &|i-j|>1 \\ s_is_js_i&=s_js_is_j &...
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Does $A$ have infinite order in $G= \langle A,B \ |\ B A B^{-1} = A^2 \rangle $?

I have a group (arising from the fundamental group of a manifold) $$G= \langle A,B \ |\ B A B^{-1} = A^2\rangle $$ and I would like to show that $A$ is an element of infinite order inside $G$. ...
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2 votes
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Show that there exists a monomorphism from $D_n$ to $S_n$.

Problem: Show that there exists a monomorphism from $D_n$ to $S_n$, $n\geq 3$. Write $D_n=\langle x,y\mid x^n=1, y^2=1, yx=x^{-1}y\rangle$. Define a map $\phi:D_n\to S_n$ such that $ \phi(x)=\begin{...
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1 vote
2 answers
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Newbie question: generating the group elements from a group presentations

I am reading about group presentation and I don't understand how to generate the group elements from the presentation when there are several relations. Take the Klein $4$ group, which, according to ...
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1 answer
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Show that $G= \langle x,y\mid x^2,xyx^{-1}y=e \rangle$ is a semidirect product of two of its subgroups

Show that $G= \langle x,y\mid x^2,xyx^{-1}y=e \rangle$ is a semidirect product of two of its subgroups my first attempt was to use the theorem: $G\cong H\rtimes K$ iff $H=$ normal $H\cap K=\{e\}$ and ...
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2 votes
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Presentation of a group from a paper of Charles Ford

The question is very small - I am trying to understand an example of a group, but its presentation looks incomplete. In a paper, Charles Ford (1970) mentions following example of a group. Let $G$ be ...
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Torsion free groups with no unique products (notation)

I am reading a paper by William Carter titled "New examples of torsion-free non-unique product groups" and saw the following group: $$P_k=\langle a,b\mid ab^{2^k}a^{-1}b^{2^k},ba^{2}b^{-1}a^{...
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6 votes
0 answers
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Show that the following group is not left orderable

I was trying to solve an exercise from a book but i got stuck in my calculation. I'm trying to show that foundamental groups of a class of orientable 3-manifold is not left orderable where by this i ...
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2 votes
1 answer
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Does the Baumslag Solitar group $B(2,3)$ contain a non-trivial element with arbitrary roots?

The Baumslag Solitar groups $B(n,m)$ are defined via the presentation $\langle a,b \mid b a^m b^{-1} = a^n \rangle$. We say that an element $g$ of a group $G$ has an $n$-th root, if the equation $g = ...
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1 vote
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Elementary divisors of group defined by relations

I've found this exercise. Determine the elementary divisors of the abelian group in two generators $\{a,b\}$ with the relation $3a=4b$. I don't see how to solve this. I believe that there should be an ...
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4 votes
2 answers
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Infinite and non-abelian fundamental group

$\require{AMScd}$ I ran into trouble while trying to answer this question. I am trying to prove the following: Suppose $U_1,U_2$ and $U_3 := U_1 \cap U_2$ are open, path-connected subsets of $X = U_1 ...
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3 votes
1 answer
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Proving an Isomorphism via Generators

Prove that it is sufficient to map the generators and their relationship onto each other in order to show that we have an Isomorophism between two Groups. Say we have $\langle g,h\mid g^3=1,h^2=1,hg=...
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Show the following presentation of the free abelian group of rank n

The free group 𝐹𝑆 consists of all reduced words that can be built from members of 𝑆 and formal inverses of members of 𝑆. Show the following presentation of the free abelian group of rank $n$: $$\...
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2 votes
1 answer
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Prove that order of $D_{2n}$ is exactly $2n$

I have been reading Dummit & Foote, and I got stuck on exercise 1 in ch 1.1, where the authors asked to show that a group with general dihedral group presentation has order exactly $2n$. To be ...
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Group representations and generators

could someone elaborate on the following: A matrix representation of a group $G$ is a homomorphism $R:G\rightarrow GL_n$. If a group $G$ is given by $$\langle x_1,\dots,x_n\mid r_1,\dots,r_k\rangle$$ ...
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7 votes
2 answers
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What are relations of $S_4$, if generators are $a=(12)$, $b=(1234)$?

I think, relations could be $a^2=e$, $b^4=e$, $ababab=e$, $b^2 a b^2 a b = b a b^2 a$. By two first relations and $b^3 = ababa$ (it is result of our relations), we can present any element that is ...
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1 vote
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If $n$ is odd prove that $D_{2n} \cong \langle a^n \rangle \times \langle a^2,b\rangle.$

Let $$D_{n}=\langle a,b\mid abab,b^2,a^n \rangle$$be the $n$th dihedral group. If $n$ is odd prove that $D_{2n} \cong \langle a^n \rangle \times \langle a^2,b\rangle.$ My solution: First of all $|D_{...
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  • 1,551
4 votes
1 answer
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Presentation of Grothendieck-Witt group $GW(\mathbb{F})$ in terms of generators and relations.

Let $\mathbb{F}$ be a field, which for the sake of this discussion, is such that char $\mathbb{F} \neq 2$. By Corollary 9.4 in Scharlau's Quadratic and Hermitian Forms, the Grothendieck-Witt group $GW(...
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1 vote
1 answer
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How to prove that $\varphi$ is an injective homomorphism and why it is surjective?

Here is the question I want to solve: Prove that the subgroup of $SL_2(\mathbb F_3)$ generated by $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 \\ 1 & -1 \...
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3 votes
1 answer
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Commutators in the infinite dihedral group

I started to learn about the infinite dihedral group $D_\infty=\langle x,y\mid x^2=y^2=1\rangle.$ Can you tell me some references about this group? I need to know about it as specifically as possible. ...
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1 vote
1 answer
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Why is a free factor of a group malnormal in that group?

For a subgroup $K \leq H$, we say $K$ is a free factor of $H$ if $H$ can be written as the free product $K * C$ for some $C \leq H$, i.e. if we have the presentations $K = \langle S_K \mid R_K \rangle$...
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