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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Surjections from free groups

I am stuck on the following: How do i go about finding surjections from the free group of rank 2 $\mathbb{F}_2$ = $\mathbb{ Z∗Z}$, to the finite group of two element $\mathbb{Z}_2$. Also,how would ...
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2answers
32 views

System of generators and surjective homomorphism

Let $G$ and $G'$ be groups and $\varphi:G\to G'$ a group homomorphism and $(g_s)_{s\in S}$ an indexed collection of elements of $G$ and is also system of generators of $G$. If $\varphi$ is surjective ...
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A simple question about free group

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. I have asked a question several days ago: Is $\mathbb{F}_{2}$ a subgroup of $\mathbb{F}_{3}$? ...
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Existence of Unique Homomorphism Implies Generating Set

The following question is taken from "An Invitation to General Algebra and Universal Constructions", p. 23 Ex. 2.1.2 (available online here). Let $G$ be a group and let $\{a,b,c\}\subseteq G$ such ...
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Prove that $A \times B \cong C * D$ is not possible

Prove that there is no group $G$ such that it has non-trivial expansion simultaneous in direct product and free product, i.e. $G = A \times B = C * D$. Check my attempt please. Let's take an ...
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29 views

Homomorphism between free group and symmetric group

Let $G:=S_5$ with elements $g_1=(12345)$ and $g_2=(12)$ and let $\varphi:F_2\to G$ be the unique homomorphism satisfying $\varphi(x_i)=g_i$ for $i=1,2$ Questions $\bullet$ Is ...
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1answer
23 views

Descending central series of free groups

What is the general term $\Gamma_k(F_n)/\Gamma_{k+1}(F_n)$, where $F_n$ is the free group of $n$ generators and $\Gamma_k(F_n)$ is the $k$th term in the descending central series of $F_n$?
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1answer
22 views

Finding free subgroups thanks to Lie algebras

Let $f : F \to G$ be a homomorphism from a free group $F$ to a group $G$. I heard that, in order to verify whether or not $f$ is one-to-one, it is possible to associate a Lie algebra $E_0^*(H)$ to any ...
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1answer
66 views

finite simple groups and free groups

It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? ...
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The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$

Show that: The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$. I've tried to apply the first isomorphism theorem to this by defining the ...
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center of free product of groups

Show the following: Let $G_1, G_2$ be groups and denote by $G_1*G_2$ their free product. Then center of $G_1*G_2$ contains only the neutral element. Have you some nice references about free ...
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35 views

Free Group Norms

Hello everyone, I'm trying to solve this problem, but I'm stuck... i don't quite understand the definition of the norm, If you guys can give me a better explanation, I would appreciate it, Thanks
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Count how many “free words” of a certain length reduce to the identity

Let $F_n$ be the free group with $n$ generators $g_1,\ldots,g_n$. I'm trying to settle the following: Question. For a fixed even integer $m$, is there a systematic way to count how many words ...
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2answers
38 views

Free Group generated by S is actually generated by S.

Consider the definition of free groups via the universal property: Definition. We say that the group $F$ is the free group generated by the set $S$ if there's a map $f:S\to F$ such that whenever ...
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36 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
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Which Words Are Part of a Free Basis of $F_n$?

Start with a free group on $n$ generators, $F=\langle a_1,\ldots, a_n\rangle$. If I write a word, $w$, in these generators, is there an efficient algorithm to determine whether or not $w$ can be ...
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348 views

Words in the Category of Sets

I was wondering about free objects in different categories and the "words" in those categories. I think I have a generally good grasp on the idea, but I started to think about stranger free objects ...
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Commutator subgroup of rank-2 free group is not finitely generated.

I'm having trouble with this exercise: Let $G$ be the free group generated by $a$ and $b$. Prove that the commutator subgroup $G'$ is not finitely generated. I found a suggestion that says to ...
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1answer
52 views

Free group on two generators and commutators. Why it's enough to add the relation ab=ba?

I've looked through lots of question on this topics, but I cannot find what I want to prove: I've seen in a lots of exercises sheets that the abelianization of a free group with two generators (let's ...
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1answer
35 views

How do I prove that a presentation of the free product is this? How does this given proof make sense?

Rotman - Introduction to the theory of groups p.390 Let $\{A_i:i\in I\}$ be a family of groups and let a presentation of $A_i$ be $(X_i|\Delta_i)$, where the sets $\{X_i:i\in I\}$ are pairwise ...
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105 views

Epimorphisms from a free group onto a free group

Let $f:F_n\to F_m$ be an epimorphism ($n\geq m$). Then it is true that there is a basis $X=X_1\sqcup X_2$ in $F_n$ such that $f$ maps $\langle X_1\rangle$ isomorphically onto $F_m$, and maps $X_2$ to ...
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1answer
48 views

Free groups of rank greater than 2

I'm trying show that a free group of rank $\ge2$ is non abelian, but I have no idea to prove this. Any suggestions?
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Total ordering on the free group

The free groups can be totally (bi-)ordered. This paper shows how to do it (page 4). In short, you embed the group in multiplicative structure of the ring of power series in non-commuting variables, ...
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Equation in Free Group [closed]

Let $F$ be a free group and $a \in F$. Assume that for any natural $n>1$ the equation $x^n=a$ has solutions (that is, $a$ is infinitely divisible). Show that $a=1$.
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Generating sets of the free group $F_k$ on $k$ generators [duplicate]

Is it true that the free group $F_k$ on $k<\infty$ generators requires at least $k$ elements to generate. I.e. does every set which generates $F_k$ have cardinality at least $k$?
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Finitely generated group $G$ such that $G\cong G*G$ must be trivial

So, I need to show that a finitely generated group isomorphic to the free product of two copies of itself (obviously thinking every factor as being generated by diferent letters) must be trivial. I ...
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1answer
35 views

Question about particular words in the free group on three generators

In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further ...
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$F/H^{\prime}$ is a torsion free group [closed]

Let $F$ be a free group and $H$ be a normal subgroup of $F$. I want to show that $F/H^{\prime}$ is a torsion free group where $H^{\prime}=[H,H]$.
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1answer
62 views

Element of Infinite Order of Finitely Presented Group

Let $G$ be finitely presented group with $n$ generators and $r$ relations where $n>r$. I want to show that $G$ has an element with infinite order. My attempt: Assume that $F_n$ is free group with ...
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1answer
50 views

Are the two inverses in the free group same?

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains ...
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Rank of a free group

I am trying to know whether the following result is true. Let $F$ be a free group with a basis $X$, and let $X'=\{xF':x\in X\}$, where $F'$ is the commutator subgroup of $F$. Then, $|X|=|X'|$. ...
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1answer
65 views

Semidirect products as (amalgamated) free product

It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?
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47 views

$\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable

I read that $\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable (ie. for every finitely generated subgroup $H$ and $g \notin H$, there exists a finite index subgroup $K$ such that $H \subset ...
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50 views

Commuting Elements in a Free Product of Cyclic Groups

In the free group with two generators $F_2\cong\mathbb Z *\mathbb Z$ ($*$ denotes the free product), if two elements $a$ and $b$ commute, then there exists an element $w\in F_2$ such that $\langle ...
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Pairs of $2\times 2$ matrices generating free groups.

The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, ...
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1answer
69 views

Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
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68 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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1answer
36 views

Random walk on free group on two elements

Let $F_2$ be the free group on two elements, generated by $\{a, b\}$. We perform a random walk on $F_2$, starting at the identity element $e$ and uniformly at random selecting one of ...
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1answer
54 views

The action of free group on line

Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?
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57 views

the center of amalgamated product of free groups

Let $G_1$, $G_2$, $H$ be free groups, $K=G_1*_H G_2$ is the amalgamated product of free groups, then is center of $K$ trivial? Thanks in advance.
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Reduced Words of Length $l$

How many reduced words are there of length l the free groups of rank $r$? Moreover I want to know about the number of cyclically reduced words? I think $r(r-1)^{l-1}$ is the answer for first ...
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1answer
40 views

Finitely generated groups without the minimal condition on subgroups

Definition A group $G$ is said to satisfies the minimal condition on subgroups iff every descending chain of subgroups stops after a finite number of steps. I know Novikon and Adjan proved that the ...
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73 views

An old problem in Group Theory

Definition A group $G$ is said to satisfies the minimal condition on subgroups iff every descending chain of subgroups stops after a finite number of steps. I know this was an unsolved problem in ...
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100 views

Fundamental group of a connected graph

Is this a legit way to prove that a fundamental group of a connected graph $\Gamma$ is a free group? Without using quotient and homotopy extension property from Hatcher's "Algebraic Topology": Take ...
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1answer
47 views

Show that the group $\langle a, b, c | ab = bac \rangle$ is free

I'm trying to use Tietze transformations tranforms it to the group $\langle a, b, c \rangle$ - is this the right thing to do?
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Free groups: $u^mv^k = v^ku^m \implies uv = vu$

I'm trying to show that if $F_n$ is a free group of rank $n$ and $u,v \in F_n$ and $m, k >0$ then $u^mv^k = v^ku^m \implies uv = vu$. I can't seem to do it by manipulating the equation $u^mv^k = ...
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If $A$ is a set of generators for the free group $F(X)$ then $|A| \geqslant |X|$.

I'm currently revising my course Geometric Group Theory - my notes say that if $A$ is a set of generators for the free group $F(X)$ then $|A| \geqslant |X|$ because otherwise there are at most ...
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1answer
36 views

Free abelian group of fractional ideals

This question is from Ch.2 of Frohlich and Taylor's Algebraic Number Theory, page 42. Let $R$ be a Dedekind domain, $I_R$ the multiplicative group of fractional $R$-ideals. There is an isomorphism of ...
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129 views

Free group of finite rank: subgroup of finite index

This is a well-known result, but I can't find a proof of it, without using topology. Let $m\geq2$ be an integer. Then the free group of rank $2$ contains the free group of rank $m$ as a finite-index ...
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a word is zero exponent sum in free group

I read a book about free groups, it says a word is zero exponent sum, but it wasn't defined before. So what is a word which is zero exponent sum?