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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3answers
48 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 ...
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1answer
48 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 ...
1
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1answer
33 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
104 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
44 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
29 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
73 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
22 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
48 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 ...
1
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1answer
27 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 ...
2
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0answers
34 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
61 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 ...
1
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1answer
49 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
votes
2answers
59 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
60 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?
2
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1answer
46 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
votes
1answer
44 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
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, ...
3
votes
1answer
56 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 ...
0
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0answers
67 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 ...
1
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1answer
32 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 ...
0
votes
1answer
51 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?
3
votes
1answer
50 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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0answers
17 views

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 ...
1
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1answer
38 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 ...
7
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1answer
71 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 ...
2
votes
1answer
78 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 ...
2
votes
1answer
45 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?
0
votes
1answer
28 views

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 = ...
1
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2answers
29 views

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 ...
0
votes
1answer
34 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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0answers
112 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 ...
2
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1answer
17 views

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?
2
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4answers
56 views

Are quotients of chain groups $C_n(X)/C_n(A)$ still free?

Suppose you have a topological space $X$ and a subspace $A$. Their chain complexes are made up of free abelian groups $C_n(X)$ and $C_n(A)$ are the free abelian groups on the $n$-simplexes on $X$ and ...
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
54 views

Cokernels in the category of free abelian groups

My question is if there are Cokernels in the category of free abelian groups. The answer is yes in the case of finitely generated free abelian groups since one has the structure theorem of finitely ...
3
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0answers
40 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
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2answers
79 views

Confusion related to definitions involving free groups

From Wikipedia ...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different ...
0
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1answer
62 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
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) ...
0
votes
1answer
29 views

The Relationship Between Generators, Relations and Group Homomorphisms

Let $G = \left\langle g_1, \ldots , g_n \mid R_1\right\rangle$ and $H = \left\langle h_1, \ldots , h_n \mid R_2 \right\rangle$. Suppose there's a mapping $f$ s.t. $f(g_i) = h_i$. Can we then merely ...
1
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0answers
50 views

Verifying $G*H$ Has Trivial Center and Elements of Infinite Order

Hypothesis: Let $G \ne H$ denote two non-trivial groups. Goal: Show that $G * H$ has a trivial center (hence is non-abelian) and contains an element of infinite order. Is my attempted proof below ...
3
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0answers
36 views

Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)

Hypothesis: Let $G$, $H$, $G'$, and $H'$ be cyclic groups of orders $m$, $n$, $m'$, and $n'$ respectively. Goal: Show that if $G * H$ is isomorphic to $G' * H'$ then $m = m'$ and $n=n'$ or else $m ...
0
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1answer
57 views

Notion of group generators

I'm asking myself what the meaning of a statement like the following is: Let $G$ be a group and $T_i \subseteq G$ be a family of subgroups of $G$ indexed by a possibly infinite set $I$. Now let ...
3
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1answer
47 views

A multiplicative subgroup of rational numbers

Let $\Bbb Q^+$ be the set of positive rational numbers and $K=\{\sqrt {t^2 +s^2}:t, s\in \Bbb Q^+\}$. It is easy to see that $K$ forms a multiplicative group since $$ ...
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0answers
30 views

Isomorphism between free pro-$p$-groups induced by isomorphism on abelianizations

Let $G$, $H$ be free pro-$p$-groups, where $p$ denotes a rational prime. I want to show that a canonical homomorphism $$G \to H$$ is in fact an isomorphism. The next step is a reduction step to ...
0
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0answers
24 views

Defect of a quasimorphism

Suppose we have a free group $F_2$ on 2 generators (say, $g_1$ and $g_2$) and and element $f=f_1 ... f_k \in F_2$ such that each $f_i$ can be one of $\{g_1,g_1^{-1},g_2,g_2^{-1}\}$ and of course ...
0
votes
2answers
55 views

Free group and normal subgroups with corresponding schreier representatives.

Let $\mathbb{F}$ be a free group on $a$ and $b$. Let $N$ be the normal subgroup of $\mathbb{F}$ generated by $a^2,b^3$ and $(ab)^2$. Similarly $H$ is the normal subgroup generated by $a^2,b^3$ and ...
2
votes
1answer
87 views

Freely generated groups and homomorphisms

Let $G$ be a group and $\{x_1,x_2,\ldots,x_n\}$ a set of its elements, such that for any group $F$ and any set $\{y_1,y_2,\ldots,y_n\}$ of elements of $F$ there is one and only one homomorphism $h: G ...
3
votes
0answers
89 views

Reducing word algorithm on the free group construction

Let $X$ be a non-empty set. There exists sets $X'$ and $\varepsilon$ such that $\varepsilon \notin X \cup X'$, $X' \cap X = \varnothing$ and $|X|=|X'|$, ie, $X$ and $X'$ have the same cardinality. ...