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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Uniqueness of induced functions on reduced free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, that is, $F_n$ modulo the relation that $[x_i,x_i^g]=1$ for all $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
4
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1answer
51 views

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$. I've been trying to prove this given the definition of a free group $F$: given group $F$ and subset $X\subseteq F$, $F$ is free ...
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0answers
73 views

Categorical proof that subgroups of free groups are free?

Is there a categorical proof that the subgroups of free groups are free? Also that abelian subgroups of free abelian groups are free.
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36 views

Two homomorphisms on the reduced free group

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, which is defined in the following two equivalent ways: $K_n=F_n/I$, where $I$ is the normal ...
5
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1answer
62 views

Proving the 'letters' of a free group generate the group

A group $F$ is free over a set $X$ if there exists an injection $\sigma: X \to F$ such that for any function $\alpha: X \to G$ to any group $G$ there exists a unique homomorphism $\phi : F \to G$ such ...
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3answers
47 views

subgroup is saturated iff it is a direct summand

Let $A$ be a free abelian group of finite rank. Call a subgroup $B \le A$ saturated if for all $a \in A$ and positive integers $n$ such that $na \in B$, the element $a$ belongs to $B$. Call a subgroup ...
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45 views

Does a finite product of commutators of the form $[a,b]$ lie in $A∩B$?

1) If $G = A★B$ be the free product of two groups $A$ and $B$, then $A$ and $B$ need not be normal to $A★B$ necessarily, right? My Aim: I need to know if finite product of commutators of the form ...
3
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1answer
163 views

Free abelian group, matrix representation.

Let $n$ be a positive integer and let $A$ be a free abelian group and let $\{e_1,\dots,e_n\}$ be a basis of $A$. Let $B$ be a subgroup generated by $\{v_1,\dots,v_n\}$ and $M=(m_{ij})$ be a matrix ...
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0answers
37 views

Co-rank of a group with $a^2b^2c^2=1$

How do I calculate the co-rank of a group $$G=\langle a,b,c,\dots,z\mid a^2b^2c^2\dots z^2=1\rangle,$$ that is, finitely generated group with one relation? By co-rank, I mean the maximum rank of a ...
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1answer
39 views

Proving a homomorphism $\varphi: F \to G$ with F a free group and $\phi: G \to H$ a surjective homomorphism and $\psi: F \to H$ a homomorphism.

Let $\phi: G \to H$ be a surjective homomorphism and let $\psi: F \to H$ be a homomorphism with $F$ a free group with basis X. Prove that there exists a homomorphism $\varphi: F \to G$ such that $\phi ...
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0answers
83 views

free group with 2 generators (two matrices) [closed]

Let $\alpha$ be complex number such that $| \alpha | > 1$. Show that $\left(\begin{array}{cc}1&0\\\alpha&1\end{array}\right) $ and ...
1
vote
2answers
71 views

An example of free group

Let $\alpha : \mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $\alpha(x)=x+2$ and $\beta:\mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $ \beta(x)=x/(2x+1)$. Show that the ...
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0answers
31 views

Rreferences for free groups

I have done free groups. I studied it from Rotman two semesters back. But this semester I am doing combinatorial group theory and obviously it starts with free groups. I have to revise Free groups but ...
3
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0answers
79 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
2
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1answer
64 views

Generators of free groups

Ive been reading an introduction on free groups and have come across some difficulty if i have the free group $\mathbb{Z*Z}$ , then what would the generators of this group be? i am confused as ...
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1answer
61 views

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 elements $\mathbb{Z}_2$. Also, how would ...
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2answers
42 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 ...
2
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2answers
66 views

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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2answers
20 views

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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0answers
41 views

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 ...
0
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1answer
34 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 ...
0
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1answer
27 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
23 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 ...
2
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1answer
70 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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0answers
44 views

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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0answers
30 views

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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1answer
39 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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0answers
35 views

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 ...
2
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2answers
41 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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37 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 ...
4
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1answer
50 views

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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2answers
357 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 ...
3
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3answers
171 views

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 ...
0
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1answer
59 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
38 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 ...
5
votes
1answer
107 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
53 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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0answers
42 views

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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2answers
76 views

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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1answer
24 views

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 $k$?
3
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1answer
52 views

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
37 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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0answers
36 views

$F/H^{\prime}$ 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 $H^{\prime}=[H,H]$.
2
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1answer
65 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
52 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 ...
2
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2answers
65 views

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, $|X|=|X'|$. ...
3
votes
1answer
69 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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1answer
50 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 ...
2
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1answer
58 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 ...
5
votes
3answers
85 views

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, ...