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

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0answers
44 views

How can I identify the following group of three generators

Let us consider the group $G= \langle a,b,c \mid aba=bab, aca=cac \rangle$. How can I determine the group identical with this group?
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2answers
47 views

Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, ...
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1answer
23 views

Von Dyck groups that are conjugated.

Let us consider the Von Dyck groups $$ D(a,b,c)=\langle x,y,z\mid x^{a}=y^{b}=z^{c}=xyz=1\rangle $$ and $$ D(a'.b',c')=\langle x,y,z\mid x^{a'}=y^{b'}=z^{c'}=xyz=1\rangle. $$ Suppose $$ ...
1
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0answers
34 views

Proof check: a group $G$ with presentation $[a,b\mid ab=e]$ is isomorphic to $\mathbb{Z}$

Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$. The presentation rewritten in terms of $a$ is ...
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0answers
17 views

About finitely generated torsion-free nilpotent groups

The only finitely generated torsion-free nilpotent group I know is the Heisenberg group: $$H= \left\{ \left(\begin{array}{ccc} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1 ...
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0answers
22 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
2
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0answers
64 views

Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
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1answer
63 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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1answer
46 views

For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong G$.

I am reading Allen Hatcher's Algebraic Topology, and am trying to understand the proof to corollary 1.28: For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong ...
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1answer
33 views

By mapping the generators $s_{i}$ into $S_{n}$ appropriately, find a well-defined epimorphism $\theta :G_{n}\rightarrow S_{n}$ .

So $G_{n}$ is the group with presentation $\left<s_{1},...,s_{n-1}\mid s_{i}^{2}=1, s_{i}s_{j}=s_{j}s_{i} \text{ if } \left | i-j \right |\geq 2, s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j} \text{ if } \left ...
0
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0answers
12 views

A function vanishing on the subgroup generated by relations defines a linear function.

I am reading Basic Algebraic Geometry 1 by Shafarevich (3rd edition) and I couldn't understand the following portion on pg 222: Namely, in Section 5.2 we defined the module of differentials ...
1
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0answers
34 views

Presentation of two Groups

Assume that a group $G$ has a presentation $\langle X \mid R \rangle$ and a group $H$ has a presentation $\langle X \mid S \rangle$. If $R \subseteq S$, the $H$ is isomorphic to a quotient of $G$. In ...
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0answers
51 views

Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the ...
1
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1answer
32 views

Expository reference on group presentations

I'm looking for expository papers, small books or chapters on the topic of group presentations. I have familiarity with basic abstract algebra (groups, rings, modules, some finite field theory from ...
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0answers
8 views

presentation of the symmetric group via transpositions fixing one element

Consider the symmetric group $S_n$. If we use the most popular set of generators $\sigma_1, \sigma_2,\cdots,\sigma_{n-1}$ with $sigma_i$ being the transposition $(i \, i+1)$, it is well known that ...
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4answers
81 views

Say we have a presentation of a finite group. Does adding additional relations to the presentation always decrease the size of the group?

It seems like it should, but I'm not sure how to prove it. EDIT: I'm talking about nontrivial new relations here, i. e. ones that do not follow directly from the old ones.
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1answer
65 views

Are these groups solvable?

I am thinking of Baumslag-Solitar groups of type $BS(1,m)=\langle a,b \mid bab^{-1} = a^m\rangle$ as a prototype. We can think of them as follows: Start with an infinite cyclic group $\langle ...
2
votes
1answer
50 views

What is the presentation of $\mathbb{Z}_n$

I am given $\langle a,b | ab=ba, b^6=1 \rangle$ and I am supposed to compute the group that has this presentation. After racking my brains for a long time, the only thing I can come up with is ...
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1answer
49 views

Why can the relators be omitted?

$$< c^2,d^3,b^2c,ca^2,bda^3d^2,ab^3a^3b^3,b^3db^3d^2a^3 >$$ is a presentation of the group $SL(2,3)$. According to GAP, the relators $c^2$ and $ab^3a^3b^3$ can be omitted, in other words, ...
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0answers
86 views

Proof of every finite group is finitely presented.

I'm reading the proof that every finite group is finitely presented from Dummit's Abstract Algebra, but there's a part that I don't understand. In the proof below, what are the elements $\tilde{g_i}$? ...
2
votes
1answer
50 views

Modify a Dehn presentation

Suppose you have a Dehn presentation $\langle X \mid R \rangle$ of (say not the free group) a hyperbolic group. Has there been some work done on changing this presentation, e.g. adding a relation ...
1
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1answer
65 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
1
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1answer
46 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
6
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1answer
110 views

On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
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2answers
37 views

Abstract finite group vs. finite group

What is the difference between the definitions of abstract finite group and finite group? I have some exposure to the finite group theory. But I came to know about abstract finite group from ...
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0answers
165 views

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...
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0answers
6 views

Presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$

I would like to determine the presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$. My effort: The presentation of the symmetric group $S_{N}$ is as follows. $\langle s_1, \ldots, s_{N-1} | (s_i ...
4
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1answer
280 views

Is a group defined by its generator set and relations?

I'm learning about generators from Dummit and Foote. They call this a presentation of the dihedral group: $$D_{2n} = \left< r,s\,|\, r^n=s^2=1,\, rs=sr^{-1}\right>$$ Does this type of ...
2
votes
1answer
46 views

Is this group simple?

I have this group presentation $G = \langle a,b | ba = a^{-1}b\rangle$. I'm wondering if this group is solvable or not. I tried to show that this group is simple, because if it is, then it can not be ...
0
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1answer
32 views

Prove that the presentation of a cyclic group of infinite order is $\langle a\mid \rangle$

Essentially, i need this for a third year mathematics project and originally i thought i just needed to have something like this: The group with this presentation is explicitly realized by the set of ...
0
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1answer
84 views

Fundamental group of sphere with 2 handles with 2 mobius bands.

Is it real to calculate?? We have a sphere with with 2 handles and with 2 glued mobius bands (red on picture). So, i think we need to use Van Kampen Theorem 2 times. 1) $ X = X_1 \cup X_2 $ , ...
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2answers
55 views

Relations in Group Presentation

In an introduction to abstract algebra, I was recently introduced to the idea of presenting a group - minimally, a group is just a set of generators along with a set of relations amongst the ...
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2answers
79 views

Presentation of Groups

I have troubles to solve this kind of exercises. For example: Let $$G_1=\langle x,y |x^3=y^4=1\rangle,~~~G_2=\langle x,y |x^6=y^6=(xy)^3=1\rangle. $$ I want to check that $G_1$ is an infinite ...
3
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1answer
62 views

What is the most general group possible?

I have read that free groups are the "most general" groups given some generators (from Wikipedia): The construction of a free product is similar in spirit to the construction of a free group (the ...
4
votes
3answers
194 views

Getting the wrong order of a finitely presented group

Let $G=\langle x, y \mid x^4=y^3=1, y^{-1}xy=x^{-1}\rangle$. What is $G$? I started by taking $$y^2=y^{-1}xy y^{-1}x^{-1}= (y^{-1}xy) ...
3
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0answers
41 views

Presentation of the special linear group $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$

I'm trying to show that $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$. For this, I defined a homomorphism $f$ from $G$ into $SL_2(\mathbb{Z})$ by $$f(a)=\begin{bmatrix}0& ...
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1answer
60 views

Isomorphism between groups with same presentation

So say I have two groups $G$ and $H$. We let $G = \langle X \mid R\rangle$ and $H = \langle Y \mid R'\rangle$ where $|X| = |Y|$ and $R$ and $R'$ are the "same" relations. (For example if $(ab)^c = e$ ...
4
votes
4answers
122 views

Upper bound for order of finite group given relations

Say I have a group with the following presentation: $$ G = \langle a,b \mid a^2 = b^3 = (ab)^3 = e \rangle $$ During a conversation someone had mentioned that the order for $G$ must be less than or ...
2
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1answer
38 views

Relation between $\langle X\cup Y|R\cup S\rangle$ and $\langle X|R\rangle,\langle Y|S\rangle$

Let $X$ and $Y$ be disjoint sets and $\langle X|R\rangle$ be the group given by the generators $X$ and relations $R$. Similarly for $\langle Y|S\rangle$. Is there a simple relation between $\langle ...
2
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1answer
26 views

$\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
0
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0answers
51 views

Unique homomorphism from a group defined by generators and relations to $S_3$.

Let $G = \langle a,b; a^2b^{-3} = 1 \rangle$ and consider $(1 2), (123) \in S_3$. I need to show there is a unique homomorphism from $G$ to $S_3$ which sends $a$ to $(12)$ and $b$ to $(123)$. I'm not ...
0
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1answer
52 views

Prove a group has a presentation (revisit)

I was actually asking the same question in here. However, the give answer didn't satisfy me. For the reader's convenience I'll re-write the post as follows: We define the following operation on the ...
2
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1answer
97 views

Presentation of Abelianization of a group

Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...
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0answers
38 views

Finite subgroups of $PSL(2,R)$

I know $PSL(2,R)$ is $SL(2,R)/SZ(2,R)$ and it is a simple group, but I do not have single clue how to get on finding its group presentation. How can I find its presentation and also I am looking for ...
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0answers
29 views

When is a finitely presented positively admissible group uniformly reachable?

This is a follow-up from this question. For a finitely presented group $G=⟨S|R⟩$, a positive element g∈G is an element of G that can be written as a finite product of elements of $S$ only. A ...
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votes
3answers
135 views

Is a finite index subgroup of a finitely presented subgroup finitely presented?

I do know Schreier's theorem, which states that a finite index subgroup of a finitely generated group is finitely generated. Other than this, I have no reason to suspect a positive answer to my ...
2
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0answers
63 views

How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
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votes
4answers
509 views

Does a Conjugacy Class always contain an element and its inverse?

The definition of a conjugate element We say that $x$ is conjugate to $y$ in $G$ if $y = g^{-1}xg $ for some $g \in G$ Now for the group $G=Q_8$ , we have the group presentation $$Q_8 = ...
6
votes
1answer
149 views

Is the minimum number of relations in a free product, the sum of the minimum number of relations in the free factors?

Say $\rho(G)$ is the minimum number of relations required to present the group $G$. Is $\rho(A*B)= \rho(A)+\rho(B)$? What can be said about $\rho(A*B)$? A while ago I was thinking about $C_3*C_4$, ...
2
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0answers
77 views

What is the group for the following presentation?

This group has four generators $a, b, c$ and $d$. Generators satisfy the following relations: $a$ commutes with $b$ and $c$ commutes with $d$. $a^2=b^2=1$ and $c^{m_1}=d^{m_2}=1$ for some integers ...