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Questions tagged [group-presentation]

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

8
votes
1answer
541 views

Is this a group? If so, what group is it?

I have the following group (at least, I think it's a group) generated by $\langle a,b,c \rangle$ where the operation $\cdot$ obeys the following rules: $a^2=b^2=c^2=1$ (where $1$ is the identity). $\...
0
votes
0answers
19 views

Determining the Cyclic Decomposition of a Finitely Generated Abelian Group given a set of relations.

I am given a group $G=\text{Span}(w,x,y,z)$ with relations defined by $$\begin{bmatrix}0&0&1&3\\-2&1&1&3\\-2&4&1&3\\0&-3&1&5\end{bmatrix}\begin{bmatrix}...
5
votes
1answer
130 views

Identifying the group $G = \langle x,y \space | \space xy=yx, x^4=y^2 \rangle$ from the given presentation [duplicate]

I'm trying to solve a problem in my textbook which asks me to identify the groups $G_1 = \langle x,y \space | \space x^3y=y^2x^2=x^2y\rangle$ and $G_2 = \langle x,y \space | \space xy=yx, x^4=y^2 \...
1
vote
1answer
39 views

Proving undecidability of group isomorphism problem from an unsolvable word problem

From The Princeton Companion to Mathematics, IV Branches of Mathematics, pages 126-127: Suppose that $\Gamma = \langle A | R \rangle$ is a finitely presented group with an unsolvable word problem, ...
-1
votes
1answer
49 views

Is it true that $\pi_1(M)=\langle a\mid aaa,aa^{-1} \rangle \cong \Bbb Z_3$? [closed]

Is it true that $\pi_1(M)=\langle a\mid aaa,aa^{-1} \rangle \cong \Bbb Z_3$? This is the fundamental group of a three dimensional manifold determined by its Heegaard diagram. I'd appreciate your ...
3
votes
1answer
58 views

Automorphisms “killing” and group

I ran into the following concept in passing here. Let $G$ be a group and let $\phi$ be an automorphism of $G$. Let $P$ be a presentation for $G$ with $X$ the set of generators in $P$. Form a new ...
2
votes
0answers
48 views

What does Sylow theory have to say about group presentations?

What does Sylow theory have to say about group presentations? Of the books on combinatorial-group-theory I have looked in so far, the following do not contain any reference to Sylow's Theorems: ...
1
vote
1answer
43 views

If $g^5 = h^7$ in a free group, then $g$ and $h$ are in a cyclic subgroup

Let $F$ be a free group, $g,h \in F$ with $g^5 = h^7$. Then I want to show these are in a cyclic subgroup. The strategy I'm trying and failing with is to show that $g$ and $h$ commute, then they are ...
7
votes
2answers
139 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
2
votes
3answers
53 views

Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
1
vote
1answer
56 views

Prove or disprove that $G = \langle{x,y\;|\; x^3, y^3, x^{13}, [x,y]=1}\rangle$ is trivial.

Prove or disprove that $G = \langle x,y \mid [x,y]=x^3=y^3=x^{13}=1 \rangle$ is trivial. So the fact that $x^3=x^{13}$ means that the order of $x$ divides both $3$ and $13$, thus $|x|=1$ and so x is ...
10
votes
1answer
185 views

Is a HNN extension of a virtually torsion-free group virtually torsion-free?

Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free group. Let $H,K<G$ be isomorphic (finite index) subgroups of $G$ and let $\varphi:H\rightarrow K$ be an isomorphism. ...
2
votes
1answer
37 views

If we add more relations to a presentation will it always form a quotient group?

Specifically, if I have a presentation $\left<G|R\right>$, and I look at the presentation $\left<G|R,R_1\right>$ is always true that $$\left<G|R,R_1\right>\cong\left<G|R\right>...
2
votes
1answer
68 views

Presentations of subgroups of S6 as permutations

Computer science professor, self-taught abstract algebraist. Beginner with GAP and SAGE. Can someone show me the quickest way to, when given the description of a subgroup of S6, obtain its ...
3
votes
1answer
60 views

Finding the order of $\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle.$

Im new at abstract algebra stuff and im wondering whats the technique to prove this kind of stuff. Question: Let $G=\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle$, prove that $|G|=16 $ and find all ...
3
votes
1answer
30 views

Focal subgroup is normal

When $H$ is a subgroup of $G$, we can define the focal subgroup of $H$ as $$H^\ast:=\langle h^{-1}h'\mid h,h'\in H, h'=h^g, g\in G\rangle.$$ I'm confused about the proposition "$H^\ast$ is a normal ...
1
vote
0answers
24 views

Number of words of length N that reduce to the identity in a specific Coxeter group

Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G = \langle g_1,...g_k | (g_i)^2 = e, (g_ig_j)^3 = e \rangle$. How many words of length $N$ simplify to the ...
-2
votes
1answer
63 views

$\pi_1(M)=\langle a,b|b^{-1} b^{-1},a \rangle \cong \Bbb Z?$ [closed]

$\pi_1(M)=\langle a,b|b^{-1} b^{-1},a \rangle \cong \Bbb Z?$ Why? Thanks in advance!
-2
votes
2answers
67 views

Why is $\langle a,b|bab^{-1} a^{-1},a a \rangle$ isomorphic to $\Bbb Z_2 \oplus \Bbb Z$? [closed]

Why is $\langle a,b|bab^{-1} a^{-1},a a \rangle$ isomorphic to $\Bbb Z_2 \oplus \Bbb Z$?
2
votes
3answers
89 views

Proving $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$

I'm trying to prove that $G := \langle a, b, c \mid abc^{-1}a^{-1}, bcb \rangle$ is not isomorphic to $H := \langle a, b \rangle$. If they are isomorphic, then their abelianizations $G/[G, G] = \...
1
vote
0answers
35 views

What is a simply presented group?

I have some background in commutative ring theory. At the moment I am going through factorization theory of integral domains. I found out that it is a conjecture, that every Abelian group is the ...
3
votes
0answers
38 views

Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
3
votes
0answers
58 views

How to classify the sets $M$ by their structures?

In this post, we construct a set of matrices with the following properties Given $M$ comprised of $n\times n$ matrices, which satisfies $I_n \in M$ and $0_{n} \not\in M$ If $A,B \in M$, ...
1
vote
0answers
51 views

How can I show that $D_{2n} \cong C_n \rtimes C_2 $

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
0
votes
1answer
54 views

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 ...
2
votes
1answer
45 views

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 (...
0
votes
1answer
28 views

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 ...
3
votes
1answer
46 views

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\...
4
votes
1answer
72 views

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 ...
1
vote
0answers
18 views

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 $\widehat{G_p} \simeq \mathbb{Z}_p$...
2
votes
0answers
25 views

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 ...
2
votes
1answer
84 views

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 ...
0
votes
0answers
27 views

$p$-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 ...
4
votes
3answers
104 views

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=...
3
votes
1answer
47 views

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 ...
1
vote
2answers
35 views

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 ...
3
votes
1answer
70 views

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 $i^4 = 1$?
2
votes
1answer
49 views

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 ...
4
votes
1answer
48 views

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?
1
vote
0answers
61 views

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?
1
vote
1answer
67 views

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^...
1
vote
1answer
53 views

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 ...
2
votes
2answers
88 views

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: $21600, 21602, 21604$?...
1
vote
2answers
78 views

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 $...
4
votes
0answers
83 views

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\...
18
votes
0answers
196 views

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\...
2
votes
2answers
58 views

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 $21$. ...
1
vote
1answer
97 views

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 $2\times A_5$ ...
1
vote
3answers
47 views

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 ...
0
votes
1answer
51 views

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 ...