Questions tagged [combinatorial-group-theory]

Use this tag for questions about free groups and presentations of a group by generators and relations.

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2
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54 views

Why this group is isomorphic to $F_\infty$?

On a study on convering spaces, I have get a subgroup of the free group $F_2$ with the generators $a$ and $b$ s.t. $H=\{a^{m_1} b^{n_1}a^{m_2} b^{n_2}\cdots a^{m_k} b^{n_k}; n_1+n_2+\cdots +n_k=0\}$ ...
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1answer
48 views

Why does the dicyclic group have exactly one involution?

This is Exercise 12.2(a) of Roman's "Fundamentals of Group Theory: An Advanced Approach". According to this search, it is new to MSE. The Details: Roman defines, on page 350 ibid., the ...
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1answer
39 views

Show that the Abelianized group $G=\langle t_1,\dots,t_n;t_{1}^{2}t_{2}^{2}\dots t_{n}^{2}\rangle$ is $\Bbb{Z}^{n-1}\oplus\Bbb{Z}/2\Bbb{Z}.$ [closed]

Show that the Abelianized group $G=\langle t_1,\dots,t_n; t_{1}^{2}t_{2}^{2}\dots t_{n}^{2}\rangle$ is $\mathbb{Z}^{n-1} \oplus \mathbb{Z}/2\mathbb{Z} $. I don't have any idea how to solve this. In ...
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Group theory presentation of group $\Bbb Z \oplus \Bbb Z$ [closed]

Show that $\langle x, y\mid[x, y] = 1 \rangle$ is a presentation for $\Bbb Z \oplus \Bbb Z$.
3
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1answer
58 views

For any primitive element $a$ in a free group of rank two we have $a^k ba^l=b$ only if $(k,l)=(0,0)$ provided $b\not\in \langle a\rangle$

Problem 1: Let $F$ be a free group of rank at least two, and $a, b\in F$ be two non-trivial elements with $b\not \in \langle a\rangle$. Suppose, $a\neq x^n$ for any $x\in F$ and any $n\geq 2$, i.e. $...
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2answers
111 views

Show that two finite two generator groups are isomorphic

I was faced by a question that I can't solve. Any help would be great! Let $A$ and $B$ be groups with the following properties: \begin{cases} |A| = 9 \cdot 3=27\\ A = \left<a,b\right> \\ a^{...
4
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1answer
105 views

Why the group $\langle x,y\mid x^3, y^3, yxyxy\rangle$ is not trivial?

This comes from Artin Second Edition, page 219. Artin defined $G=\langle x,y\mid x^3, y^3, yxyxy\rangle$ , and uses the Todd-Coxeter Algorithm to show that the subgroup $h=\langle y\rangle$ has ...
2
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1answer
46 views

Word problem in the braid group

The geodesic length of elements can be defined as the length of a minimal path from 1 to w in the Cayley graph of G. This length is dependent on the particular generating set X. The Word Problem is ...
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0answers
41 views

Largest Hopfian quotient

Let $\Gamma$ be a group, say finitely generated if it helps. Does $\Gamma$ admit a largest Hopfian quotient? That is, does there exist a Hopfian quotient $H$ of $\Gamma$, such that every surjective ...
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Minimal presentation length of a universal finitely-presented group?

It is a rather well known fact, that there exist universal finitely presented groups (finitely presented groups, that contain all other finitely presented groups as subgroups). It is a rather direct ...
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1answer
43 views

Does every set of generators of a finite group contain a minimal set of generators? [closed]

Suppose that $G$ is a finite group that can be generated from $n$ elements $g_1,...,g_n\in G$. Now, if $S\subseteq G$ is another set of generators of $G$, can I always find a subset $S'\subseteq S$ of ...
2
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1answer
28 views

A question in Reidemeister–Schreier rewriting process

I am reading about Reidemeister–Schreier rewriting process and I have the following question. After we find a set of generators for the subgroup we find the set of its relators which are of the form $...
0
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1answer
59 views

How do I write $ba$ in the form $a^i b^j\ $?

Let $p$ be a prime number and $q \in \Bbb N$ be such that $q\ |\ (p-1).$ Let $u \in \Bbb Z_p^*$ be an element of order $q.$ Now consider the group $G$ defined by $$G = \left \langle a, b\ \big |\ a^p =...
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2answers
62 views

On 'proving' that $\langle x,y_1,\ldots,y_n | [x, y_1 \cdots y_n] = 1 \rangle \simeq \mathbb Z^2$.

As the title says, I want to prove that $G:= \langle x,y_1,\ldots,y_n \mid [x, y_1 \cdots y_n] = 1 \rangle \simeq \mathbb Z^2$ algebraically. Edit: well, this is embarrassing. That is not what I want, ...
0
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1answer
71 views

what is the group $\langle a,b,c,d,e\mid cde=1\rangle$?

As title, I'm doing a problem in algeraic topology, and met this group, I need to figure out what it is. I met similar group such as $\langle a,b,c\mid c=1\rangle $ and $\langle a,b,c,d\mid cd=1\...
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Group $G =\langle a, b\rangle$ with $a^{p^m}=b^{p^r}=1$ and $b^{-1}ab=a^n$ has commuting subgroups.

Here $n^{p^r}\equiv 1\pmod{p^m}$ and $p$ is prime. We say $A, B\leq G$ commute if $AB =BA$. I've shown that commuting subgroups is equivalent to $xy = y^tx^s$ for any $x, y\in G$ and some $t, s\in \...
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1answer
27 views

Why do certain relatively free groups have perfect commutator subgroups?

I am trying to understand an argument in a group theory, which is not my strong suit (I mostly work with semigroups and rarely delve into the actual structure of groups). Let $\boldsymbol{\mathcal{V}}$...
9
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1answer
60 views

Does every finite group $G$ have a set of generators such that the sum of the orders of the generators is less than or equal to $|G|$?

Does every finite group $G$ have a set of generators such that the sum of the orders of the generators is less than or equal to $|G|$? This is surely true but I am failing to see why. This is easily ...
2
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1answer
28 views

On normal forms of group elements

I was reading about finitely presented groups. Some finitely presented groups (like the braid groups) have normal forms, which means that every element of that group has a unique representation of ...
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1answer
62 views

Prove that some group is infinite based on its presentation.

Suppose $G$ is a group with presentation $$G = \langle a, b, c\mid a^2=b^2=c^2=(ab)^3=(bc)^3=(ca)^3=1\rangle.$$ I want to show that this group is of infinite order. I think since the element $abc$ is ...
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2answers
47 views

Need help understanding what is a free factor of a free group.

Let $F_n$ be the free group with $n$ generators $\{ x_1 , \dots , x_n \}$. From my understanding, we say that a subgroup $K \leq F_n$ is a free factor of $F_n$ if there exists a subgroup $H \leq F_n$ ...
3
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1answer
33 views

Cardinality of the equivalence class of a given braid word?

The symmetric group $S_n$ has presentation $$S_n = \{ s_1,...,s_{n-1} | s_is_{i+1}s_i=s_{i+1}s_is_{i+1}, \text{ } s_i^2=1 , \text{ and } s_is_j = s_js_i \text{ for } |i-j| \geq 2\}$$ If we take away ...
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1answer
62 views

Prove $s(n-k-1)=k(k-r-1)$ [duplicate]

I don’t know where i saw this question but any help would be appreciated. The question is: In a crowd with $n$ people , each person have $k$ friends, also every two people who are friends have $r$ ...
2
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1answer
42 views

Equivalent condition of the following group presentation to give a group of order $mn$; the other side

Let $G$ be a group (not necessary abelian) generated by $a, b$ that has a group presentation: $$\langle a, b \mid a^m = 1, b^n = 1, ba = a^rb\rangle. $$In my question Equivalent condition of the ...
2
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1answer
44 views

Equivalent condition of the following group presentation to give a group of order $mn$

Let $G$ be a group (not necessary abelian) generated by $a, b$ that has a group presentation: $$\langle a, b \mid a^m = 1, b^n = 1, ba = a^rb\rangle. $$ I have to prove that $G$ is of order $mn$ if ...
2
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1answer
136 views

Cayley graphs obtained for different generating elements with the same orders in a finite group

When we consider the finite group $\mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime, $p>2$, the set with the pair of elements $\{(0,1), (1,0)\}$ can generate the group. Moreover, a set $\{(1,...
8
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1answer
58 views

How to find free generators of a subgroup generated by conjugates of certain elements

Reviewing old qualifying exams in algebraic topology, I see a lot of the following type of question: let $F$ be the free group on generators $a$ and $b$. Describe a set of free generators of the ...
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1answer
28 views

Generators of free groups are really generators

The free group $F_X$ generated by the set $X$ is characterized by a map $i:X\longrightarrow F_X$ which has the universal property that for any group $G$ and any map $f:X\longrightarrow G$, there is ...
4
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2answers
119 views

When two elements of a group generate the same subgroup

Let $G$ be a group and $x,y$ two elements from $G$. Suppose that $\langle xy \rangle = \langle yx \rangle$ , i.e. they generate the same subgroup of $G$. I want to find a counterexample in which $xy \...
3
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1answer
48 views

Group acting on trees with relations $abc=1$ and $cba=1$

Suppose $G$ is a finitely presented group that acts on a tree $T$ by isometries, and let $a,b,c\in G$ with relations $abc=1$ and $cba=1$. If two of $a,b,c$ are hyperbolic, does this imply the third ...
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0answers
40 views

Elements of $\langle S\rangle $ for $\emptyset \neq S\subseteq G.$

Let $G$ be a group and let $\emptyset \neq S\subseteq G.$ Prove that if $G$ is a multiplicative group, then $$\begin{align}\langle S\rangle &= \{a_1^{k_1} a_2^{k_2}\cdots a_s^{k_s} : s\geq 0, a_i \...
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1answer
125 views

Generators increasing the word metric

Let $G$ be an infinite group with a finite generating set $T$ which is symmetric ($T=T^{-1}$) and let $|\cdot|$ be the corresponding word metric, i.e. $|g|$ is the minimal number of (not necessarily ...
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0answers
36 views

Show that two presentations are isomorphic

It is known that if there is a non-abelian group of order $pq$, then it must be the case that $q\mid p-1$ and this group is isomorphic to $\langle a,b:a^p=b^q=1,ab=ba^u\rangle $ wherein $u$ is of ...
5
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1answer
173 views

Do you know this finitely presented group on two generators?

I computed using Sage the fundamental group of some topological space and got the infinite group $$\langle a, b\mid aba^{-1}ba\rangle.$$ By the change of variables $x=b^{-1}$ and $y=a$, it can also be ...
1
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1answer
39 views

Determining the conjugacy class of the non-abelian group of order $pq$

I'm trying to find the conjugacy classes of non-abelian group of order $pq$ $(p>q)$. It is known that this group is uniquely expressed as $$\begin{align*} G&=F_{p,q}\\ &=\langle a,b:a^p=b^q=...
3
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0answers
27 views

Why does no non-trivial reduced words in $H_1 \backslash \{e\} \sqcup H_2 \backslash \{e\}$ imply that $\left< H_1, H_2 \right> = H_1 \ast H_2$?

I am currently studying combinatorial group theory and am trying to prove the ping pong lemma. Many of the proofs I have come across seem to use a result of the following flavour. Let $G$ be a group ...
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1answer
37 views

Showing two group presentations are isomorphic to known groups [closed]

I'm trying to help a friend through his Maths degree and he gave my a question that I just can't remember how to do. Show $$\langle x,y | y^5=x^2=[x,y]=1\rangle \ \text{and}\ \langle u, w | u^{-1} wu=...
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2answers
518 views

Can the map sending a presentation to its group be considered as a functor?

It is well-known that the functor $Grp \to Set$ sending a group $G$ to its underlying set $UG$ has a left adjoint, the functor $Set\to Grp$ sending a set $X$ to the free group $FX$. I wonder whether ...
0
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1answer
46 views

malnormal subgroup of amalgamated free product

Consider the amalgamated free product $\Gamma = K\ast_{H\simeq H'} L$. Let $A$ be a malnormal subgroup of K i.e, for all $k\in K\setminus A$, $k^{-1}Ak \cap A ={1}$. Is A malnormal in $\Gamma$? I was ...
4
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2answers
113 views

Subgroups of free groups which avoid conjugacy classes

Let $G = (\mathbb Z/2\mathbb Z)^{\ast m}$ be a free product of some groups of order $2$. Let $\alpha_1,\ldots,\alpha_m$ be the generators. Can I find a free, nonabelian subgroup of $G$ that has no ...
0
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1answer
30 views

Normal closure of a finite set

I am studying a proof that states that in every finitely presented residually finite group $G=\langle X \mid R \rangle $ the word problem is solvable. At some point it states that we can enumerate all ...
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1answer
42 views

Free group of a free group.

Let $G=\frac{F}{R}$ be the presentation of a group $G$. Clearly here $F$ is free group. Can we obtain a free group $\mathbb{F}$ of the group $F$. Is $\mathbb{F}\cong F?$ What will be the presentation ...
5
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1answer
80 views

Are solvable groups Howson?

A finitely generated group $G$ has the Howson property if the intersection of any two finitely generated subgroups is again finitely generated. (Finitely generated) free groups, nilpotent, and ...
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0answers
56 views

Presentation of finite abelian groups

If $G$ a non cyclic abelian group, then $$G\cong \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\cdots\times \mathbb{Z}_{n_k}.$$ Can we give any presentation of this? My attempt- $$G=\langle g_1g_2\cdots g_k:...
2
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0answers
77 views

Dihedral Group generated by reflection

Good evening, I need some help, with the following task. I know that there are two reflections $t,t'$ with $\langle t, t' \rangle = D_n$. Now I should tell what I can say generally for two reflections ...
0
votes
1answer
63 views

Determine a group's order by Universal Property (Mapping Property)

For a group represented as $\langle x,y\mid x^4,y^5,xyx^{-1}y\rangle$, how to determine its precise order? I guess I may need to use the universal property, but how to construct functions to determine ...
0
votes
1answer
58 views

How to find the relation for $Q_8 ?$

Problem taken from dummit and Foote section $1.5$ Find a set of generators and relations for $Q_8$. My attempt : The quaternion group $Q_8$ is defined by $ Q_8=\{1,-1,i,-i,j,-j,k,-k\} $ Therefore ...
2
votes
1answer
41 views

Using an infinite number of Tietze transformations

I have a group presentation $G\cong\langle R|S\rangle$ which I am willing to reduce to $G\cong\langle S'|R'\rangle$ by making use of Tietze transformations. In my case, I am only using the following ...
1
vote
2answers
74 views

$\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}$ is an index two subgroup of $\mathbb{Z}\ast\mathbb{Z}$

I want to prove $\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}$ is an index two subgroup of $\mathbb{Z}\ast\mathbb{Z}$. Can I use covering map to prove this? Any ideas and suggestions are greatly appreciated....
1
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1answer
37 views

Group presentation, central subgroup

In a proof that I am reading there is the following statement. For a group $G$ with presentation $$ G= \langle \gamma_1 ,\gamma _2,\gamma _3 ,c \mid \gamma _1^pc^{-1} =\gamma_2^qc^{-1}= \gamma _3^r c^{...

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