2
votes
1answer
64 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
3
votes
2answers
59 views

A finitely presented group $G$ is given by a presentation $\langle S,R\rangle$, where $R$ is finite. Show that $S$ is finite.

We were asked to prove this theorem in an exercise. This is what I have thus far: Suppose $S$ were infinite. Denote the set of symbols of $S$ that do not occur in any relation by $S'$. Then the free ...
1
vote
2answers
103 views

Group presentations - again

My question is about finding presentations for finite groups. It's along similar lines to my earlier question -- but is subtly different! The earlier question is here Group presentations Let's take ...
3
votes
1answer
67 views

$\mathbb{C}[x,y\,|\,x^m=1,y^n=1]\cong\mathbb{C}[z\,|\,z^{mn}=1]$ as complex algebras?

If $G$ is any finite abelian group and $K$ an algebraically closed field with $|G|\neq 0$ in $K$, then the group algebra $K[G]\cong M_{n_1}(K)\times\cdots\times M_{n_k}(K)$ by Maschke's and ...
2
votes
3answers
97 views

A group presentation for $\mathbb{Z}_2\times \mathbb{Z}_2$

I know that the only groups of order 4 are $\mathbb{Z}_2\times \mathbb{Z}_2$ and $\mathbb{Z}_4$ up to isomorphisms. And I also know that the group presentation of $\mathbb{Z}_4$ is $\left ( a:a^4=1 ...
2
votes
1answer
89 views

Presentation of Heisenberg Group $\mathbb{H}$ over the field $\mathbb{F}$

Let $\mathbb{F}$ denote the finite field. Denote $\mathbb{H}_{\mathbb{F}}=\left\{ \left( {\begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} } ...
2
votes
1answer
55 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 ...
1
vote
0answers
64 views

If $G/N$ and $N$ are finitely presented, then $G$ is finitely presented.

Let $G$ be a group and $N\triangleleft G$. Show that if $G/N$ and $N$ are finitely presented, then $G$ is finitely presented. Worked on this problem for about 2 hours before we all threw in the ...
1
vote
0answers
84 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
2
votes
0answers
37 views

The order of the group $\langle a, b| w^3, w\in\langle a, b\rangle\rangle$ for $w$ being any word.

I came across this group mentioned in passing as finite. Does anyone know the order of the group, and where I can find a proof of this quantity? Replacing $3$ with $n$, does this problem have a name ...
4
votes
1answer
89 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 ...
5
votes
2answers
138 views

Finitely generated group which is not finitely presented

Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented? I was looking for a finitely presented group with a central ...
3
votes
0answers
160 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
4
votes
1answer
77 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 : ...
3
votes
3answers
115 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 ...
1
vote
1answer
98 views

Group presentations and subgroups

How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$? $\bf Edit$: Given that $|G|=12$.
5
votes
6answers
218 views

A presentation of a group of order 12

Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$. I tried to let $d=ab\Rightarrow G=\langle d,c\mid ...
5
votes
2answers
145 views

notations of generators and relations

I need to understand the following about generators and relations notations: Is $\langle a,b \mid a^kb^l\rangle =\langle a,b\mid a^k=b^l\rangle =\langle a,b\mid a^k,b^l\rangle$? Is $ \langle a,b\mid ...
0
votes
1answer
150 views

presentation of a group

Let $G=F( x,y)$ be a free group with two generators, assume that $H\leq G$ where $H=\langle xyxy^{-1}\rangle $, let $N$ be the normal closure of $H$. Is it true that $G/N$ has a presentation $\langle ...
3
votes
2answers
75 views

Unprovability of $i^2 =1$ from $\langle i \mid i^4 =1\rangle$ and similar problems

This question is related to Can I derive $i^2 \neq 1$ from a presentation $\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle$ of Quaternion group $Q$? I know I'm going too far but let me just ask... ...
1
vote
2answers
104 views

tensor product and direct product of algebra presentations

Let $R$ be a commutative unital ring and $R\langle x_i\mid f_j\rangle$ denote a unital $R$-algebra presentation. Q1: What is the presentation of $R\langle x_i\mid f_k\rangle\otimes R\langle y_j\mid ...
1
vote
2answers
145 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
2
votes
2answers
248 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
1answer
64 views

Proving that an element in an algebra presentation is nonzero

Let $F$ be a field, $F\langle x,y\rangle$ the free $F$-algebra on two generators (polynomials in two noncommuting variables), $A= F\langle x,y\,|\, xy\!=\!1\rangle= F\langle x,y\rangle/\langle\langle ...
1
vote
1answer
185 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 ...
1
vote
3answers
165 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
3answers
109 views

Does there exist a finite group with the following presentation?

Let $G$ be a finite group (with only two generators and $m=n$) presented as $$ G = \langle a, b : a^m = b^n = (W(a,b))^p= \ldots\text{other-such-relations}\ldots= 1 \rangle $$ where $m,n,p>1$ , ...
2
votes
1answer
138 views

What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or ...
0
votes
1answer
97 views

If $a^{3^3}=b^9=1$ for generators $a,b$ of $G$, can we conclude that $G$ is a $3$-group?

I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$. Are these informations sufficient to affirm that the group is a $3$-group? Adding the relation $b^{-1}ab=a^4$, can we ...
6
votes
1answer
483 views

Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
2
votes
2answers
206 views

$\langle X|\emptyset\rangle\ncong\langle X|R\rangle$ for finitely presented groups (exercise in Massey)

Let $:F_n$ denote the free group of rank $n$. How can I solve the exercise 7.6.3.(b), page 234, in Massey's Algebraic Topology? I'm guessing there's been made a mistake and (b) actually reads "If ...
2
votes
1answer
197 views

Quotient of free group with normal subgroup

I'm trying to make up an example of a quotient of a free group to check if I understand quotients properly. I do for the usual cases but I've not seen free groups before. So here I go: Let $F = ...
2
votes
2answers
374 views

free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$

I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication ...
0
votes
1answer
187 views

the formulation of Nielsen-Schreier theorem: every subgroup of a free group is free

I just got a little confused reading the formulation on wiki. Let $F_X$ denote the free group on the set $X$ and let the symbol $\leq$ denote "is subgroup of". From what I know, the theorem reads: ...