The free-groups tag has no wiki summary.

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### Bases of free groups

Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...

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### Generators of Sylow subgroups

Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...

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### Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...

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### Schreier's formula and descending chains

For a group $G$ we denote by $d(G)$ the cardinality of a smallest set of generators.
A finitely generated group $G$ is said to satisfy Schreier's formula if for every subgroup $H \subseteq G$ of ...

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### (Quadratic) equation in free group?

I know almost nothing about this field, so my following question may be stupid.
We know in a free group F, if $ax^{-1}a^{-1}x=1$, then $x=t^{k}, a=t^{l}$ for some t.
Now consider a more general ...

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### Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?

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### Normal Subgroup Growth

Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number ...

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### Schreier's index formula

A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...

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### intersection growth of free profinite groups

Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$).
My questions are:
can I ...

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### finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$.
Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that ...

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### Number of subgroups of a given index of a free group

Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of
rank $n$ have?
In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups.
I think I ...

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### “Concretely” writing down elements in a free profinite group

Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...

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### Dynamics of virtual automorphisms of free group

The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.
Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the ...

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### Algebraically Independent Numbers and Affine Linear Maps

Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in ...

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### Generating certain subgroups of free groups via primitive elements

Let $F_n$ be the free group on $n\geq 2$ generators and let $H < F_n$ be a finite index normal subgroup. Let $P\subset F$ be the subset consisting of primitive elements (an element of $F_n$ is ...

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### First-order axiomatization of free groups

Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)?
Is there ...

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### Elements of minimal length in normal closures of elements in free groups

Let $F_n$ be a free group of rank $n$. Let $w\in F_n$ be cyclically reduced.
What can be said about the element(s) of minimal length from the $\textit{ncl}(w)$ (normal closure of $w$ in $F_n$)? ...

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### Stable commutator length of elements in free groups.

http://arxiv.org/pdf/math/0611889v4.pdf (page 13)
In the above paper by Danny Calegari he says that the result $\text{scl}(g) \geq 1/2$ (i.e. a stable commutator length $\text{scl}(g) := ...

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### Automorphism classes of the free group

As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F_2$ ...

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### Conjugacy classes of elements in free groups. One-variable equations.

First of all, wasn't sure what could be a good title for this question. If mods think of a better name, pls feel free to change it...
Let $F$ be a (non-abelian) free group of finite rank, a vector ...

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### A question on normal closures of elements in free groups.

Let $F$ be a free group of finite rank, and $p, b \in F$, where $b$ is a root element (i.e. not a proper power).
I have a case where $p^{n_k} = V_{n_k}^{-1}b^{-1} V_{n_k} \cdot U_{n_k}^{-1}b ...

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### How to solve this one-variable equation in a free group?

Let $F_n$ be a free group, $u, v \in F_n$, where $[u,v] \neq 1 $. Trying to show that the following equation does not have a solution in $F_n$ :
$$ [v,x] = [u,v] .$$
Any ideas are appreciated. I ...

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### Commutators in free groups

Let $F_m$ be the free group with $m$ generators $S:=\{x_1,\dots, x_m\}$. I am interested in the following quantity
$$
F(n):=\frac{|\{w\in [F_m,F_m]: \|w\|_S\leq n\}|}{|B_S(n)|}
$$
where ...

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### Commutator length modulo finite index subgroups

We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group.
Given an element $g$ in the commutator subgroup of the free group $G=F_2$ ...

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### Explicit path in the unitary group of a $C^*$-algebra

For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...

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### Direct proof that a group is Hopfian

Consider the group $G=\langle x_1,x_2,x_3|x_1^2,x_2^2,x_3^2\rangle$. Using a slightly modified version of S. Ivanov's proof here that free groups are residually finite, I can show that this group is ...

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### Non-generating sets in a free group.

Let $F_n$ be the free group generated by $x_i$, for $1\leq i\leq n$. Let $a_i$ be some elements of $F_n$, also for $1\leq i\leq n$. Is there a nice way to tell when the list $\{a_i^{-1}x_ia_i\}$ does ...

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### What are the relations between conjugates and commutators?

The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are:
$a^b= ...

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### Primitive subwords in a free group of rank 2

I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following:
Suppose $w\in F_2$ is a primitive word whose length is big enough. Then ...

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### Which polynomials are Fricke polynomials ?

Let me recall the definition which seems the most standard of Fricke polynomials.
Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique ...

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### Lefschetz numbers for homomorphisms of free groups

Let $G = F_X$ be the free group on a finite set $X$, and $\phi\colon G\to G$ a group homomorphism. Consider the number
$$ \sum_{x\in X} (\text{number of occurrences of the generator $x$ in the word ...

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### Generating set for abelianization of “mod $p$ commutator subgroup” of a free group

Let $F_n$ be a free group on $n$ letters, and fix some prime $p \geq 2$. Define
$$K_{n,p}=\text{ker}(F_n \rightarrow H_1(F_n;\mathbb{Z}/p))$$
and
$$V_{n,p} = H_1(K_{n,p};\mathbb{Q}).$$
For $x \in ...

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### Algorithm for image of a free group homomorphism

Let $G$ and $H$ be finitely generated free groups, and let $f:G\to H$ be a homomorphism specified by giving the images of the generators of $G$.
Is there an algorithm which takes such an $f$ and a ...

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### How to determine free generators of a closed subgroup of a free pro-$p$-group ?

If $F$ is a free discrete group, then any subgroup $H$ of $F$ is free: this is the well-known theorem of Nielsen-Schreier. Moreover, there is a well-known algorithm, the Nielsen-Schreier
method that ...

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### A metabelian quotient of a free group

I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me.
Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. ...