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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94
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2answers
3k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
26
votes
10answers
3k views

I don't understand what a “free group” is!

My lecture note glosses over it really, introduces it and says "well it intuitively makes sense" but I say, nope it doesn't. Free groups on generators $x_1,...,x_m,x_1^{-1},...,x_m^{-1}$ is a ...
17
votes
5answers
2k views

Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
15
votes
5answers
863 views

Prove elements generate a free group

How does one show that the elements $x^2$, $y^2$, and $xy$ have no nontrivial relations among them in the free group generated by $\{x,y\}$? This would prove that the free group $F_2$ has a subgroup ...
14
votes
1answer
355 views

When is $G \ast H$ solvable?

In a proof that the lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}$ is not finitely presented, I showed that $\mathbb{Z}_2 \ast \mathbb{Z}$ is not solvable. More precisely, one can prove that the ...
12
votes
8answers
2k views

Why the group $< x,y\mid x^2=y^2>$ is not free?

$G= \langle x,y\mid x^2=y^2\rangle $. I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
12
votes
3answers
477 views

Could the concept of “finite free groups” be possible?

Is it possible to define "finite free groups" ? could that make it easier to deal with group presentations ?
11
votes
3answers
3k views

Finding subgroups of a free group with a specific index

How many subgroups with index 2 are there of a free group on two generators? What are their generators? All I know is that the subgroups should have 2*2 + 1 - 2 = 3 generators.
11
votes
2answers
438 views

What good are free groups?

In Algebra: Chapter 0, one learns two definitions of Free Groups associating with sets. Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where ...
10
votes
1answer
414 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 ...
9
votes
4answers
565 views

Is the free group on an empty set defined?

I'm guessing that the free group on an empty set is either the trivial group or isn't defined. Some clarification would be appreciated.
9
votes
2answers
851 views

Free group as a free product

Let $G$ be a group generated by two elements $a$ and $b$. Suppose $G$ is a free group of rank 2. Is it true that $G=\langle a\rangle * \langle b\rangle$? I think the problem is that the definitions I ...
9
votes
2answers
113 views

Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
8
votes
2answers
228 views

Free groups in some classes?

I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'. Can someone please tell me what are the solvable free ...
8
votes
1answer
119 views

What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
7
votes
3answers
833 views

Quotient of two free abelian groups of the same rank is finite?

Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B ]$ (the index of $B$ in $A$) is finite. For ...
7
votes
3answers
545 views

Free group n contains subgroup of index 2

My problem is to show that any free group $F_{n}$ has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2? The subgroup needs to have ...
7
votes
3answers
951 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 ...
7
votes
1answer
179 views

Does this group presentation define a nontrivial group?

Given a presentation $$ \langle x,y,z : x^y=x^2, y^z=y^2, z^x = z^2 \rangle, $$ where $x^y$ is just the usual conjugation. Can we say for sure, whether this presentation defines a nontrivial group?
7
votes
1answer
61 views

Free subgroup of linear groups

Suppose $a,b \in GL(n,\mathbb{C})$, and $\langle a,b\rangle$ is a free group of rank $2$. Is there a way to choose a $c$ to guarantee that $\langle a,b,c\rangle$ is a free group of rank $3$?
7
votes
1answer
87 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 ...
7
votes
2answers
500 views

Group isomorphism concerning free group generated by $3$ elements.

From Jacobson's Basic Algebra I, page 70, Let $G$ be the group defined by the following relations in $FG^{(3)}$: $$x_2x_1=x_3x_1x_2, \qquad x_3x_1=x_1x_3,\qquad x_3x_2=x_2x_3.$$ Show that $G$ ...
6
votes
2answers
279 views

What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
6
votes
1answer
729 views

Power of commutator formula

A few people remember a commutator formula of the form $[a,b]^n = (a^{-1} b^{-1})^n (ab)^n c$ where $c$ is a product of only a few commutators (say $n-1$) of them. Here $a,b$ are in a (free) group ...
6
votes
3answers
406 views

The free group $F_2$ contains $F_k$

I want to prove the following: the free group $F_2$ contains the free group $F_k$ for every $k \geq 3$. I am wondering whether the following line of reasoning is correct or not: Suppose that $\lbrace ...
6
votes
3answers
505 views

Alternative “functorial” proof of Nielsen-Schreier?

There are two proofs of Nielsen-Schreier that I know of. The theorem states that every subgroup of a free group is free. The first proof uses topology and covering space theory and is rather elegant. ...
6
votes
1answer
161 views

Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
6
votes
2answers
491 views

Lower central series of a free group

Consider the element $w=x^2yx^{-1}y^{-1}x^{-1}yxy^{-1}x^{-1}$ of the free group $F_2=\langle x,y\rangle$. By considering the image of this element under the abelianization map (equivalently, by adding ...
6
votes
1answer
72 views

Automorphisms of free groups

Suppose $U$ is a subgroup of finite index in the free group on $k$ generators $F_k$. Suppose $\sigma$ is an automorphism of $F_k$ such that $\sigma|_U = \text{id}$, then must $\sigma = \text{id}$?
6
votes
1answer
134 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.
6
votes
1answer
138 views

Automorphism of the free group

Let $\mathbb{F}_2$ be the free group of rank $2$ with generators $a$ and $b$. I would like to build an automorphism $\varphi$ of $\mathbb{F}_2$ such that : 1) $\varphi([a,b]) = [a,b]$ 2) ...
6
votes
2answers
54 views

Free groups and derivative

Per definition a derivative on a group $G$ is a mapping $D:G\rightarrow\mathbb{Z}G$ such that $D(gh)=D(g)+gD(h)$. Now my question: uppose $G$ is a free group $F=F(X)$ with $X$ a finite set and suppose ...
5
votes
4answers
1k views

Generators of a free group

If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?
5
votes
3answers
102 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, ...
5
votes
1answer
127 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 ...
5
votes
2answers
334 views

Free group in GAP

I know that in the free group $F$ with two generators $x$ and $y$, there is some word $w \in [F,F]$ such that $xy^2=x^{-2} y^{-3} x^{-2}(xy)^5 w$. Is it possible to find $w$ using GAP?
5
votes
2answers
135 views

free group generated by polynomials

Someone recently asked me how to proof that $x+1$ and $x^3$ generate a free group. A colleague has worked out a proof. I have a vague memory that this has been studied, maybe a Monthly problem? Does ...
5
votes
1answer
1k views

Commutator subgroup of a free group

Let $F_k$ be the free group of rank $k$. If $k=2$ it is not hard to see that the set $\{[s_1^{n_1},s_2^{n_2}] \mid n_i\neq 0\}$ is a basis for $F_2'$. (Prime denotes the commutator subgroup). What ...
5
votes
1answer
90 views

Unique homomorphism, commutator subgroup.

Let $F = F(A)$ be a free group, and let $f: A \to G$ be a set-function from the set $A$ to an abelian group $G$. What is the easiest way to see that $f$ induces a unique homomorphism $F/[F, F] \to G$, ...
5
votes
1answer
82 views

Permutation Group as quotient of Free Group

I could use some help with the following question: Let $S_{n}$ be the permutation group of $\left\{ 1,...,n\right\}$ , what is the minimal $k\in\mathbb{N}$ such that $S_{n}$ is a quotient of ...
5
votes
1answer
79 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 ...
5
votes
1answer
42 views

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$. The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that ...
5
votes
1answer
36 views

A Quotient of Free Group

If $F$ is a free group on a finite set $S$, then the squares in $F$ generate a normal subgroup $N$ and $F/N$ is elementary abelian $2$-group of order $2^{|S|}$. Let $F$ be free group on infinite set ...
5
votes
1answer
48 views

How to Identify a Quotient of a Given Free Group

$\newcommand{\Z}{\mathbf Z}$ Problem. Let $G$ be the free group generated by three symbols $a, b$ and $c$, and let denote $G$ by writing $F(a, b, c)$. Let $N$ be the normal subgroup of $G$ ...
5
votes
0answers
211 views

Identifying a certain subgroup of a free group

Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ ...
4
votes
5answers
223 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
4
votes
2answers
423 views

A free group on the non-empty set $X$ is solvable iff $|X| =1$

Let $X$ be a non-empty set. Prove that $F_X$, the free group on $X$ is solvable if and only if $|X| = 1$. We can see that if $|X| = 1$, then $F_X$ is abelian, and hence solvable. However, the other ...
4
votes
1answer
197 views

Slicker construction of the free product of groups

The usual construction of the free product of groups $\{G_i\}_{i \in I}$ consists of taking the discriminated union $\coprod_{i \in I} G_i$ and taking the set of words satisfying a handful of ...
4
votes
5answers
1k views

Every nonidentity element in a free group $F$ has infinite order

I'm trying to prove that every nonidentity element in a free group $F$ has infinite order. I'm really new on free groups and I found this subject really strange I couldn't understand it very well yet, ...
4
votes
1answer
114 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.