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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43 views

Proving that free group actions are faithful.

Assume group $G$ acts on set $X$. My course notes say this: Free action is a special case of faithful action. To have faithful action it suffices that any $g \neq e$ in $G$ moves something (but not ...
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60 views

Good introduction to free groups and free products

In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products. Right now, ...
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50 views

Quotient of a free group on a set

Suppose I have a free group $F_S$ on a set $S,$ with a nonempty proper subset $T.$ If $N$ is the smallest normal subgroup of $F_S$ containing $T$, I would like to show that $F_S/N$ is also free. There'...
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33 views

When does a RAAG act on a Tree?

Show that a RAAG acts on a tree of valence 4, acting transitively on vertices, if at least one pair of vertices is not joined by an edge. I'm trying first to prove that every such RAAG $G$ acts on a ...
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Free monoid on indexed sets and arbitrary operations

In the proofwiki, there is this definition: The free commutative monoid on an indexed set X=⟨Xj:j∈J⟩ is the set M of all monomials under the standard multiplication. That is, it is the set M of all ...
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150 views

Whether a group containing a free group is also a free group

Suppose $G$ is a group generated by two elements $s,t$. Suppose $H$ is a subgroup of $G$ such that $H= \langle s^k,t^k \rangle$ is a free group where $k$ is some integer not equal to $1$. Does it ...
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37 views

Injective maps from free group on two generators

Why are there no injective continuous maps from $F_2$ - the free group on two generators - and $\mathbb{Z} \times \mathbb{Z}$? Is it because the free group is uncountable but $\mathbb{Z} \times \...
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49 views

Rank of a free group times a free abelian group.

I know that the rank (i.e. minimal number of generators) of the product $\mathbb{Z}\times F_2$, of the infinite cyclic and free group on two generators, is three, but the only argument I could quickly ...
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61 views

For $N\unlhd G$ for $\mathfrak{B}$-group $G$ with $G/N$ a free $\mathfrak{B}$-group, show $\exists H\le G$ with $G=HN$ and $H\cap N=1$

This is Exercise 2.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". In the book, maps are evaluated from left to right. According to Approach0, the exercise is new to ...
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Equivalence by automorphisms of two elements in the commutator subgroup of a free group?

I'll use the commutator notation $[x,y] = xyx^{-1}y^{-1}$. Let $F_{2n}$ be a free group with generators $x_1,...,x_{2n}$ and consider the two elements $$ a = [x_1,x_2][x_3,x_4]\cdots [x_{2n-1},x_{2n}]...
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66 views

Subsemigroup of free semigroup that is not free

Is there any example of subsemigroups of a free semigroup, for exemple of $(\mathbb{N}^*,\times)$ generated by primes, that isn't free ? I know that for free groups there is the Nielsen-Schreier ...
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Equivalence of two elements of the free group under automorphisms?

I have two elements of the form $$ w = x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} x_6^{a_6} $$ and $$ w' = x_1^{b_1} x_2^{b_2} x_3^{b_3} x_4^{b_4} x_5^{b_5} x_6^{b_6} $$ for integers $a_i$ and $...
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Group generated by the edges of a graph and its subgroup of based cycles

Starting from a graph, I can construct the free group generated by its oriented edges, then I can quotient by the relations $e\cdot \overline{e}=1$ where $e$ and $\overline{e}$ denote the same edge ...
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63 views

Free group on $\{x,y\}$ and Group $\mathbb Z\times \mathbb Z$

Let $F$ be free group on $\{x,y\}$ and Let $G=\mathbb Z\times \mathbb Z$ be a group with operation coordinatewise addition. So there exists an unique homomorphism namely: $$\varphi:F\to G\\with\\\...
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68 views

What is the free abelian group on $\mathbb{N}$?

I learnt that the free abelian group on a set $X$ is the group $(\operatorname{Hom}(X, \mathbb{Z}), +)$. Okay, this sounds all right, but I also know the famous result that $\mathbb{Z}^{\mathbb{N}}$ ...
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40 views

Decidability of whether a set of matrices generate a free group.

Is there an algorithm to determine whether a specified set of $k$ invertible $n \times n$ matrices generate a free group $F_k$? We have to make some assumption on the matrix entries: it seems ...
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46 views

Slick argument to find free group in $SO(4)$

It is well known that ${\rm SO}(3)$ contains the free group ${\rm F}_2$, so clearly ${\rm SO}(n)$ for $n>3$ does, too. However, the standard proof with ${\rm SO}(3)$ is somewhat annoying (the one ...
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Presentation of the amalgamated product of $G_1$ and $G_2$ above $H$ is $ \langle S_1,S_2\; ; \; R_1,R_2,\phi_1(s)\phi_{2}^{-1}(s),s \in S \rangle $

Here, I found the proof of presentation of the free product of groups. I wanted to show the same thing for amalgamated free product of 2 groups i.e. Show that that if $H$ is generated by $S$, and $ \...
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41 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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Example of homomorphism $\underset{\alpha \in A}{\ast} \pi_1(U_\alpha) \to \pi_1(X)$, where $U_\alpha \ni x_0$ are an open cover of $(X,x_0)$.

I’m reading a passage leading up to the Van Kampen’s theorem, and I have trouble understanding the settings. Here’s the materials. Let us consider a pointed space $\left(X, x_{0}\right)$ with an open ...
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66 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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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 ...
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55 views

Is this relationship on subcartesian products of quotient groups over the free group correct?

I am trying to check if the following relationship is correct. Let $F$ be a free group over some set $X$. Let $\alpha_1, \alpha_2, ..., \alpha_n$ be epimorphisms from $F$ to groups $H_1, H_2, ..., H_n$...
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How's the 'integer lattice' or the 'direct sum' of $\mathbb{Z} \oplus \mathbb{Z}$ not a Free group?

My question is about the direct sum $\mathbb{Z} \oplus \mathbb{Z}$ which is a Free Abelian group and not a free group. The the integer lattice, or what I think is the direct sum $\mathbb{Z} \oplus \...
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46 views

S_3 is a quotient of the free group F({x,y})

I am self-learning Algebra: Chapter 0 by Paolo Aluffi. He defined a presentation of a group $G$ as follows: So, according to my understanding, $R$ is the kernel (a normal subgroup) of the surjection $...
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29 views

How to prove that the characteristic polynomial of this specific matrix is not the power of a linear polynomial?

I was reading an algebra paper, and the problem that appeared to me is the following: The authors defined a group $G = A \rtimes \left<x\right>$, where $A$ is a finitely generated free abelian ...
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30 views

Elements of a free group commute if their powers commute

Let $F=<X>$ be a free group and $v, w \in F$ such that $v^n \cdot w^m = w^m \cdot v^n$. Show that $v\cdot w = w\cdot v$. I didn't think it'd be very hard, but it turns out I'm stuck. I've ...
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Final object in the “free group” category $\mathscr{F}^A$

The following excerpts come from the book Algebra: Chapter 0's section II.5 - Free groups. My question is about an easy exercise. First, to define the free group $F(A)$ on $A$, the author introduces ...
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44 views

Factor group of a free group

Let $F[A]$ be the free group on the generating set $A$. Let $C$ be the commutator subgroup of $F[A]$, then show that $F[A]/C$ is a free abelian group with basis $\{aC \mid a \in A\}$. It is trivial ...
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31 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 ...
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48 views

Free groups as free product of infinite cyclic groups

Let $S$ be an arbitrary set (countable or uncountable). It is clear that the free abelian group generated by $S$ is isomorphic to the direct sum $$\bigoplus_{s\in S}\mathbb{Z}.$$ Is the free group ...
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27 views

What does “subgroup of $F_n$ consisting of all words of exponent sum $k$” mean?

I am currently reading the following paper by Birman and Hilden, https://www.jstor.org/stable/1970830?casa_token=7y0vMR6x_lsAAAAA:...
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References on Free Profinite Groups

I would like to study free profinite groups (i.e., profinite completion of free groups) and I am looking for some introductory materials on this topic. Can anyone give me some recommendations?
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Free abelian group $F_i$ with free basis $B_i$

I have to prove this statement: Let free abelian group $F_i$ with free basis $B_i$ for $i=1,2$ then $F_1\cong F_2$ iff $\lvert B_1 \rvert = \lvert B_2 \rvert$. I prove it by the idea that element of ...
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71 views

Is $\bigoplus_{x \in X} \mathbb{Z}$ reflexive?

Basically the title. I am trying to figure out whether or not $Hom(Hom(\bigoplus_{x \in X} \mathbb{Z}, \mathbb{Z})) \cong \bigoplus_{x \in X} \mathbb{Z}$ with the isomorphism $i(y)(f)=f(y)$. Here is ...
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46 views

How is $\mathbb C(x)^*$ a free group?

My professor gave an example of a free group as the field of rationals over one variable $\mathbb C(x)^*$ but I do not know how. could anyone explain this to me, please?
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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 ...
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Exact sequence of 4 abelian groups, 3 of them being free

Let $G$ be a finitely generated abelian group that fits in an exact sequence of the form $$0 \to \mathbb{Z} \to \mathbb{Z}^n \stackrel{f}{\to} G \stackrel{g}{\to} \mathbb{Z}^m \to 0$$ for some $n \geq ...
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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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Free group $F(a,b)$ of rank 2 and free subgroup of rank $\infty$

Q: $F(a,b)$ is free group of rank 2, give an example (and proof it!) of free subgroup with infinity rank. My attempt: I was trying with $G=[F(a,b), F(a,b)]$ and $G=\langle {a^nb^n : n\in N}\rangle $ ...
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32 views

Show that F2 has an infinite index free subgroup of rank 2 [duplicate]

I am trying to show that F2 (the free group on 2 elements) has an infinite index, free subgroup of rank 2. As it has to be a free subgroup of rank 2 I'm guessing I need to find 2 elements in F2 that ...
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52 views

Retract of $F_2$ onto $[F_2,F_2]$

I'm trying to show that the commutator subgroup $[F_2,F_2]$ is not a retract of $F_2$. I was trying to do the proof by contradiction so I assume there is a retract $f: F_2 \to [F_2,F_2]$, then I know ...
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82 views

Exercise about Free Groups

Someone could help me with this proof contained in "A course in the theory of Groups" by Derek Robinson? Let $F$ be a free group on a subset $X$. If $Y$ is a nonempty subset of $X$, prove ...
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38 views

Determining whether two free groups are conjugates

Let $A= \langle a,b \rangle$, i.e. the free group on two generators. Also, let $B=\langle b^{-1}a^{-1}bab,b^{-1}a^2b, b^2, a^2, ab^2a^{-1}, abab^{-1}a^{-1} \rangle$, $C= \langle a^{-1}b^{-1}aba,a^{-1}...
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Finitely presenting implies finite presenting subset?

I recently had the following problem as a homework assignment. Let $S$ be a finite generating set of a group $G$. Suppose that G is finitely presenting over $S$, and let $R$ be any set of relations ...
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14 views

$\mathcal{N}_p$-words and free pro-$p$ groups

Let $\omega$ be a word such that every finitely generated pro-$p$ group $H$ satisfies $H/\overline{\omega(H)}$ nilpotent-by-finite. Choose some free pro-$p$ group $H$ generated by $x_1,...,x_d,y$. Why ...
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51 views

Covering space of Bouquet of Two circles

Given a bouquet of 2 circles $B_2$, is there a way to find a covering space of $B_2$ such that the deck transformation group of our covering space is isomorphic $F_2 / <R>$. i.e. the group $G = \...
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22 views

Free Groupoid over a finite directed chain

I'm trying to understand a free groupoid on a directed chain of n vertices. I know the objects will be the vertices of our graph and the morphisms will be concatenations of the edges in the graph and ...
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2answers
122 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 ...
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55 views

Showing that the free group of a disjoint union is isomorphic to the free product of the corresponding free groups

P. Aluffi's "Algebra: Chapter $\it 0$", exercise II.$5.8$. Still more generally, prove that $F(A\amalg B)=F(A)*F(B)$ and that $F^{ab}(A\amalg B)=F^{ab}(A)\oplus F^{ab}(B)$ for all sets $A,B$...

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