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Questions tagged [free-groups]

Should be used with the (group-theory) tag. Free groups are the free objects in the category of groups and can be classified up to isomorphism by their rank. Thus, we can talk about *the* free group of rank $n$, denoted $F_n$.

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Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\} $ and $G_s \neq \{e\}$ (where $G_i$ denotes the ...
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Example III.3.2(1) of Baumslag's “Topics in Combinatorial Group Theory”: proving $F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$ is free.

This question is a little tricky (for me, at least), since in the textbook the proof of Theorem 5: Every subgroup of a free group is free. is not yet provided (even though I've seen such proofs ...
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Finitely generated nilpotent group is isomorphic to a quotient of the free nilpotent group.

Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group ...
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348 views

Difference between “generating set” and free product?

Let $G$ and $H$ be free groups and $g \in G$. Is there a difference between $\langle H, G\rangle$ and the free product $H*G$?. In particular is $\langle H ,g \rangle = H * \langle g \rangle$?
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How do I show that the direct sum of F2 with itself is a one-ended group? [on hold]

How do I show that $\mathbb{F}_2 \oplus \mathbb{F}_2$ is a one-ended group? I can see why visually but I have no idea how to argue it mathematically.
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Is there a formula for $[F_n : V_{\{x^3\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_3(n) = [F_n : V_{\{x^3\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_3(n)$? ...
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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 $f$ ...
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30 views

Exact value of exponential growth rate depends on generating set

I am trying to solve an exercise from Clara Loh's Geometric Group Theory: An introduction. The problem uses the exponential growth rate of a finitely generated group $G$ with generating set $S$. The ...
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25 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 ...
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injective homomorphism between two free groups [duplicate]

I read the textbook and in the paragraph where the definition of a free group is introduced met the task (almost immediately after the definition). The task put me in a stupor. Build an injective ...
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Is this an example of comaximal ideals $I,J$ such that $IJ\not=I\cap J$?

Let $R$ denote the set of all finite formal sums of elements in the free group $\left<a,b\right>$ with the relation $a+b=1.$ Let $I$ be the principle ideal generated by $a$ and $J$ the principle ...
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2answers
55 views

Universal Property…The Map is not Well-defined?

How does the following proposition from this book make sense: Proposition 2.8 Let $G$ be the group defined by the presentation $(X,R)$. For any group $H$ and map of sets $\alpha : X \to H$ sending ...
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3answers
117 views

Show that the free group on three generators is a subgroup of the free group on two generators

I have been asked to show that the free group on three generators is a subgroup of the free group on two generators. The following definition has been taken from the appendix to Armstrong's $\textit{...
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Does a finitely generated group $G$, ever act on $G/N$ freely?

Say that $G$ is a finitely generated group on $k \geq 2$ generators and $N$ is a normal subgroup of $G$. I want to know if I can construct a $G$-action on $G/N$ such that the action is free. None of ...
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1answer
24 views

Right action induced by free group homomorphism

I was reading this proof by Steinberg of Nielsen-Schreier theorem and i had a doubt about proposition 1, that basically is an alternative universal property of free groups. It says: Let $X$ be a set ...
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1answer
75 views

An abelian group $G$ is free abelian if and only if satisfies the projective property

I've been reading group theory from Rotman's book "Introduction to the theory of groups" and in Chapter 10 of free abelian groups there is an exercise which am having a hard time to prove. The ...
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1answer
41 views

Subgroups of free groups question

I am just reading Allufi chapter 0. I have a specific question in regards to a comment that the book made. "By Proposition 6.9, every nontrivial subgroup of $\mathbb{Z}$ is in fact iso-morphic to $\...
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1answer
62 views

Free subgroups of $PSL(2,\mathbb{Z})$ of index 6

There are two "natural" subgroups of $PSL(2,\mathbb{Z})\cong C_2\ast C_3$ of index 6. One is the congruence subgroup $\Gamma_0(2)$ which is the kernel of the map $PSL(2,\mathbb{Z})\to PSL(2,\mathbb{Z}/...
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1answer
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Connection between Freiheitssatz and Magnus property

I am currently studying the Magnus property: Let $G$ be a group and $u, v \in G$. If the normal closures of $u$ and $v$ coincide, then $u$ is conjugate to $v$ or $v^{−1}$. I was told that this ...
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Proof that Sanov subgroup is free of rank 2

Okay so i was trying to prove that Sanov subgroup is free of rank 2 without using ping pong lemma. I'd like to prove it directly(if possible), using the universal property. So Sanov Subgroup is a ...
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1answer
30 views

Algorithm to find the n-th root in the free group

Given a word $w$ in the free group $F$, is there an algorithm to find it’s n-th root, if it exists? Thanks!
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2answers
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Solving the equation $abc=cba$ in the free group.

It is known that if two words $a,b$ commute in the free group $F$, then they are powers of the same word, i.e. $a=c^r$ and $b=c^s$, where $c\in F$ and $r,s \in \mathbb Z$. What happens if there are ...
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Subgroups of direct products of free groups

I am reading the following paper of Miller: http://researchers.ms.unimelb.edu.au/ He says that if $G= F_{1} \times F_{2}$ is a direct product of two free groups and $H$ is a subgroup of $G$, then it ...
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2answers
83 views

Free groups are residually of rank 2

Let $F_X$ denote the free group on the set $X$, and $F_n$ the free group of rank $n$. I have read that any free group is residually $F_2$, and I was trying to understand this. For any free group $F$,...
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1answer
64 views

Are free groups and free actions related?

Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the ...
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2answers
56 views

A generating set of cardinality $n$ in the free group $F_n$ is a free basis.

Let $F_n$ be the free group on $n$ letters. Let $S=\{s_1,s_2,\ldots,s_n\}$ be a set of $n$ elements of $F_n$. Is there any way to prove that $S$ is in fact a free basis for $F_n$ without using the ...
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1answer
49 views

Nielsen-Schreier formula application

So I was thinking about Nielsen-Schreier formula for the generators of a free group subgroup and i tried to apply it. Something must be wrong but I can't find the mistake. So I have $F(a,b)$ free ...
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Regular Covering of $S^1 \vee S^1$ with genus 1

The question is to prove that every finitely generated free group can be realized as a normal subgroup of the free group F(2). I want to do it by considering regular covers of $S^1\lor S^1$. For the ...
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2answers
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Is studying a free group (or other free object) equivalent to considering only the consequences of the basic axioms?

I'm trying to get a better understanding of the rationale behind free groups, and more generally free objects. This answer does a great job at explaining how various free objects are built, and I ...
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What is a transitive action of free group on 2 generators to a set with $n$ elements for any $n$?

Can we construct such an action? because then that solves this question: For every positive integer $n$, the free group of rank 2 has a subgroup of index $n$
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Free groups: Finding words vanishing in two different situations

Let $\varphi_i:G\to H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $\mathrm{Kern}(\varphi_1)\cap \mathrm{Kern}(\varphi_2)$ which are not contained in the commutator $[G,G]$. ...
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1answer
39 views

Any group can be represented as the fundamental group of a 2-dimensional topological space

I saw in the textbook affirmation that any group can be represented as the fundamental group of a $2$-dimensional topological space. Without proof. May be you can give some ideas, how can I proof that ...
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1answer
44 views

Number of homomorphism between a free group $F(S)$ and a group $G$

Let $F(S)$ be a free group with finite rank, $G$ a group with order $n$. I want to know if the universal property can help determine the number of homomorphisms $F(S)\rightarrow G$. Say $H$ is the ...
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1answer
26 views

An infinite left word in some alphabet

What is the infinite left word in some alphabet? I can understand the definition of the infinite right word -- some path in Cayley graph from $id$-element to $+\infty$ or $-\infty$. But is the ...
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1answer
18 views

Basic assertion about central extensions

I'm having trouble verifying what ought to be a relatively simple detail in a proof of Milnor's book on algebraic K-theory, in the section on universal central extensions. Here is the set up. Let $G$ ...
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1answer
74 views

The (un)decidability of the Tits Alternative for any given (suitably defined) set of groups.

Please forgive me if this question is ill-formed. I don't know much about decidability. Some Background: There are problems in combinatorial group theory that are undecidable, such as the word ...
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2answers
64 views

Can $G/2G$ and $\mathbb Z / 2\mathbb Z \times … \times \mathbb Z / 2\mathbb Z$ be not isomorphic? $G$ is a free abelian group with finite basis.

This is from Theorem 67.8 of Munkres Topology: The proof gives stronger than bijective correspondences, specifically isomorphisms between $G$ and $\mathbb Z \times ... \times \mathbb Z$ and $2G$ and $...
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1answer
40 views

How can I represent by generators and relations the set of real numbers

How can I represent $\mathbb{R^{+}}$,$\mathbb{R^{*}}$,$\mathbb{Q^{*}}$,$\mathbb{Q^{+}}$ by generators and relations on them? Like that $\langle a,b,c,\cdots\mid \cdots\rangle$?
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1answer
30 views

Generators and relation in group and *non-triviality*

Consider the following basic problem: let $G$ be a group with following generators and relations $$G=\langle x: x^2=1, x^3=1\rangle.$$ It is easy to play with generators-relations to conclude that $x=...
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1answer
34 views

Difference between $F(S)$ and $G$ when $S$ generates $G$

Let $G$ be a group generated by a set $S$. It is known that the inclusion map $i: S \to G$ can be extended to an epimorphism $I: F(S) \to G$. Is there a difference between $F(S)$ and $G$? Clearly $G ...
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1answer
25 views

$F=\langle a_1 ,…, a_n \rangle$ is a free group, show: $w\in [F,F] \Leftrightarrow \sigma_i(w)=0 \forall 1\le i \le n$

We define $\sigma_i(w): F\rightarrow \mathbb{Z}$ the number of appearances of the generator $a_i$ in $w$ when $a_i , a_i^{-1}$ cancels each other and we work with reduces word (means there are no ...
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1answer
36 views

Centraliser of a subgroup of Free Group

Let $\mathbb{F}_n$ be a free group on $n$ generators. Let $H$ be a subgroup of $\mathbb{F}_n$. When is $$C_{\mathbb{F}_n}(H)=\{g \in \mathbb{F}_n: gh=hg, \forall h \in H\}$$ the centraliser of $H$ in $...
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1answer
75 views

Computing $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$

I want to compute $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$ with respect to homomorphisms: $\varphi_1:\mathbb{Z} \ni n \longmapsto an \in \mathbb{Z}$ $\varphi_2:\mathbb{Z} \ni n \longmapsto bn \in \...
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3answers
113 views

$G=\langle a,b \mid baba^{-1}=1\rangle$ Show that $\langle a \rangle$ is infinite

Let $G=\langle a,b \mid baba^{-1}=1\rangle$. Show that the subgroup generated by $a$ is infinite. My attempt Suppose $\langle a\rangle$ is finite so $a^k = 1$ for some $k \in \mathbb{Z}$. So I ...
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1answer
70 views

What could be the elementary divisors of subgroup of $\mathbb{Z}^2$

What can be the elementary divisors of subgroup $H \le \mathbb{Z}^2$ of index $36$? I can't see what's the connection between the index and the elementary divisors? As far as I know, elementary ...
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86 views

Elementary divisors of a subgroup of $\mathbb{Z^2}$ of index $36$

The question is what might be the elementary divisors of a subgroup $H\leq \mathbb{Z^2}$ of index 36? I need to list all the possible options. My solution: The theorem states that if $A$ is an ...
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2answers
74 views

How many epimorphisms from $F_2$ to $\mathbb{Z}_5$?

How many epimorphisms are there from $F_2$ (the free group with $2$ generators) to $\mathbb{Z}_5$? If $F_2$ is generated by $2$ elements, so it has a basis of rank$2$, meaning that it is isomorphic ...
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2answers
55 views

Amount of elements of order $p^2$ [closed]

Let $p \neq 5$ be a prime number. How many elements of order $p^2$ are there in $\mathbb{Z_p} \times \mathbb{Z}_{p^5} \times \mathbb{Z}_{25}$? I have no idea how to even approach this... Any hints?
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61 views

Representing curves using elements of a free group

Assume that two piecewise smooth homotopic closed curves $\gamma_1$ and $\gamma_2$ with same endpoints, divide the plane into finite number of regions $R_0, R_1, \cdots R_n$, where $R_0$ is the unique ...
2
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3answers
59 views

Confusion on the definition of presentation of Groups

We define a group presentation to be an ordered pair denoted by $\langle S | R \rangle$ where $S$ is an arbitrary set and $R$ is an arbitrary subset of the free group $F(S)$. We call the elements of $...