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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85
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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: ...
13
votes
5answers
923 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 $X$?
13
votes
1answer
300 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
5answers
397 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 ...
11
votes
3answers
396 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 ?
9
votes
4answers
314 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
542 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 ...
8
votes
3answers
2k 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.
8
votes
2answers
255 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 ...
8
votes
2answers
216 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 ...
7
votes
3answers
325 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
1answer
70 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
297 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
3answers
375 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
2answers
50 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
1answer
376 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 ...
5
votes
3answers
81 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
2answers
129 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 ...
5
votes
3answers
205 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 ...
5
votes
1answer
90 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
125 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
117 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?
5
votes
2answers
258 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 ...
5
votes
1answer
74 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
52 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 $\sigma = \text{id}$?
5
votes
1answer
85 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) ...
5
votes
0answers
190 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
4answers
806 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?
4
votes
2answers
205 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
141 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
2answers
147 views

Is $\Bbb{6Z}$ a free group?

I'm trying to understand the concept of free groups , and from what I've learned so far , a group $G$ is called a free group , if there is a subset $S ⊂ G$ such that any element of G can be written ...
4
votes
1answer
94 views

A lemma about free groups

Let F be a finitely generated free group and $\gamma_m$ the lower central series. Why is $\gamma_m(F)/\gamma_{m+1}(F)$ torsionfree? I know it is abelian, but I couldn't find out more about it, as ...
4
votes
1answer
56 views

Proving things about the free group from the categorical definition.

I'm undertaking A Course in the Theory of Groups by Robinson and I'm looking for some guidance on some of the exercises. Specifically, I'm trying to show that a free group of rank 2 or higher has a ...
4
votes
1answer
340 views

Equation c./ commutator subgroup, direct sums, and free products (Munkres, Section 69, Exercise 1)

Let $G = G_1 \ast G_2$. Then $$G/[G,G] \simeq G_1/[G_1, G_1] \oplus G_2/[G_2,G_2].$$ This is Munkres, section 69, exercise 1; with the hint to use the extension condition for free groups and ...
4
votes
1answer
242 views

Why isn't this free product of groups abelian?

I'm trying to prove that the free group $A=A_1*A_2$, where $A_1, A_2\neq 1$ is not abelian. Following the hints below: Let $x,y\in A_1*A_2$, where $x\neq y$. Suppose now $A_1=F(S)$ and $A_2=F(T)$, ...
4
votes
0answers
66 views

Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
3
votes
5answers
102 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?
3
votes
3answers
442 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 ...
3
votes
4answers
215 views

$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: ...
3
votes
3answers
66 views

Showing a Mapping Between $\left\langle a,b \mid abab^{-1}\right\rangle$ and $\left\langle c,d \mid c^2 d^2 \right\rangle$ is Surjective

Hypothesis: Let $$ G \cong \left\langle a,b \mid abab^{-1}\right\rangle $$ $$ H \cong \left\langle c,d \mid c^2 d^2 \right\rangle $$ Let the function $f$ be defined as follows. First let $f(a) ...
3
votes
3answers
148 views

Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
3
votes
3answers
178 views

Proving that a group generated by x,y and z and a given relation is actually free

I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group. What are some usual ways of going about this ...
3
votes
1answer
47 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.
3
votes
2answers
166 views

Must a surjection $F_2\to F_2$ also be an injection?

In other words, are there words $w_1$ and $w_2$ in the free group $F_2$ such that $F_2=\langle w_1, w_2\rangle$ but which are not a free basis for $F_2$? I'm sure I'm missing a simple argument or a ...
3
votes
1answer
744 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 ...
3
votes
2answers
142 views

Question on the presentation of $(\mathbb{R}, +)$

In this question, it is shown that $(\mathbb{R}, +)$ is not a free group. But my question is: if it is not a free group, exactly what relations is it subject to? My other question is: are there ...
3
votes
2answers
140 views

Normal subgroup of automorphisms of a free group

Let $F_2=\langle X,Y\rangle$ be the free group of rank $2$ and consider $A,B,C\in Aut(F_2)$ given by: $$A(X,Y)\mapsto(YX^{-1}Y^{-1},Y^{-1})$$ $$B(X,Y)\mapsto(X^{-1},Y^{-1})$$ ...
3
votes
1answer
47 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 ...
3
votes
1answer
54 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?
3
votes
1answer
53 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 ...