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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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
41 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
43 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
34 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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3answers
91 views

What is the cardinality of free product $\mathbb{Z} * \mathbb{Z}$? [closed]

I want to know cardinality of $\mathbb{Z} * \mathbb{Z}$. Is it countable? or uncountable?
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3answers
76 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
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1answer
51 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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0answers
65 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
29 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
47 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
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1answer
44 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
14 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 ...
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1answer
33 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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1answer
69 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
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1answer
63 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
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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?
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1answer
25 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 = ...
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2answers
27 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 ...
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1answer
31 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
77 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
14 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?
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4answers
52 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 ...
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1answer
55 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 ...
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1answer
38 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 ...
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0answers
39 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
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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 ...
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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 ...
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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) ...
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1answer
24 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 ...
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0answers
40 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 ...
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0answers
34 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 ...
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1answer
56 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
45 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
25 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 ...
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0answers
21 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 ...
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2answers
54 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
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1answer
69 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
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0answers
84 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. ...
2
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2answers
50 views

Basis for singular chains group

The singular chain group $S_p(X)$ is defined as the free abelian group generated by continuous functions $T \in C( \Delta_p , X)$. What I understand this means is that we define $T' \in S_p(X)$ as ...
5
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2answers
119 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 ...
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5answers
859 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 $X$?
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1answer
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Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
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1answer
111 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?
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47 views

Subgroup of a (free) group.

Let $F$ be a group, generated by $x_1,...,x_m$ and $H$ be its subgroup such that $|F:H|=n < \infty$. How to prove that $H$ can be generated by $n(m-1)+1$ elements?
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2answers
101 views

Group presentations - again

My question is about finding presentations for finite groups. It's along similar lines to my earlier question -- but is subtly different! The earlier question is here Group presentations Let's take ...
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1answer
63 views

Does every group have a representation?

For any set $A$, we can give it an group structure and make it a free group. For example: $$\mathbb Z=<a;aa^{-1}=1>$$ Further more, we can introduce some relation on it: $$\mathbb ...
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1answer
103 views

How many normal subgroups is in a free group of rank > 1

How many normal subgroups is in a free group of rank $k>1$, if the quotient group (of the normal subgroups) isomorphic $S_3$?
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1answer
39 views

Two free basis of a free abelian group

We have a free abelian group $A(X)$, where $X$ is its free basis, and let $Y$ another free basis for $A(X)$. We know that every $g\in A(X)$ can be expressed as $g=a_1x_1+...+a_nx_n$ where the $a_i$'s ...
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1answer
100 views

Free groups contain “larger” free groups

First of all, I have read this similar question and am satisfied that the answers there prove the result I am interested in. That being said, I'm more interested in this particular approach than in ...
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5answers
378 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 ...