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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How can I show that $D_{2n}$ follows from these relations?

Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$ It should be just the dihedral group of order $2n$, the one generated ...
2
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1answer
53 views

Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
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1answer
37 views

Reference Request: Subgroup of free abelian group is free abelian

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis ...
2
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1answer
49 views

Construction of free abelian group from free group

I am reading Fraleigh's Abstract Albebra recently, and I cannot prove a statement about free abelian group: Let $F[A]$ be a free group generated by set $A$ and $C$ is the commutator subgroup of ...
3
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3answers
82 views

Is every normal subgroup of a finitely generated free group a normal closure of a finite set?

Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$. My question is, if $G$ is a free ...
7
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1answer
54 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 $3$?
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1answer
23 views

derivative free optimaization method

Currently I am working project on the derivative of free optimization methods. however, I want find practical problem that solved using this method. So, how can I get solve practical examples using ...
2
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1answer
43 views

What is the probability of 2 random matrices generate a free group?

Let A,B $\in GL(2,Z)$, then what is the probability of $<A,B>\cong F_2$? By probability, I mean the haar measure on $GL(2,Z)^2$. I already know what if we replace $\mathbb{Z}$ with ...
2
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1answer
26 views

Non-amenability of $F_2$ via $l^2$-homology

Since the free group $F_2$ of rank $2$ is non-amenable, we have that $l^2$-homology of $F_2$ vanishes in degree $0$, i.e. $H_0(F_2,l^2(F_2))=0$. This means that for any $f\in l^2(F_2)$, the equation ...
3
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0answers
38 views

Lower central series of Free group 4

I have a question. I found the following statement in some paper. Let $F$ be free group on two generator $x$ and $y$ and $F_i$ be $i$-th component of lower central series of $F$. If $F$ is generated ...
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16 views

Lower central series of a free group 3

Let $F$ be free group on two generator $x$ and $y$ and $F_i$ be $i$-th component of lower central series of $F$. If $F$ is generated mod $F_1=[F,F]$ by $b_1,...,b_n$ then $F_r$ is generated mod ...
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1answer
38 views
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103 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?
2
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1answer
74 views

Prove $G \cong \langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$

Let $D^3= D \times D \times D$ where $D = D_\infty$ where we see $D$ as the group generated by $\mathbb{Z}$ and element $0^*$ of order $2$ such that $0^*n0^*=-n$ for all $n \in \mathbb{Z}$. Letting ...
0
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1answer
50 views

Specific topological example of Nielsen-Schreier theorem

I'm assuming that the following question should be basically trivial, and that I'm just misunderstanding something basic, but some clarification would be much appreciated. There is a section in my ...
2
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1answer
69 views

Group presentation of Integers $\big(\mathbb{Z,+}\big)$

I can't understand how is it possible to represent the group $(\mathbb{Z},+)$ as follows $$\mathbb{Z} = \big<a\big>$$ with only one generator and no relations ? How can there be no relations ...
2
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0answers
42 views

Free groups of rotations of the sphere

Is the following conjecture true: If $G$ is a group of rotations of the sphere and $G$ contains two noncommuting rotations of infinite order, then $G$ has a free subgroup of rank $2$. By the Tits ...
6
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1answer
110 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 ...
3
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1answer
66 views

free groups of rotations

The question of which pairs of rotations of the sphere are independent goes back to Hausdorff, who produced such a pair a century ago. "Independent" meansĀ "are free generators of a free group". The ...
2
votes
2answers
72 views

How do i proof that that the map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is homomorpism?

I'm trying to proof that a map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is a homomorphism but i can't exactly define which is the function to show that. The map $ \varphi $ is a map with for any ...
3
votes
1answer
36 views

Prove that $A$ is a free abelian group

Suppose $a_1, \dots, a_n$ generate an abelian group $A$, and for any abelian group $B$, and any $b_1, \dots, b_n \in B$ we can find a homomorphism $\varphi: A \to B$ given by $\varphi(a_i) = b_i ~ ...
2
votes
1answer
50 views

Proof that elements of a free generating set have infinite order.

I'm trying to show that elements of a free generating set $S$ have infinite order straight from the definition of a free group being generated by $S$. The definition I'm using is that a group $F$ is ...
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2answers
50 views

Quotienting by generators in free groups

I feel like this is a simple result but have not touched algebra in a while and can't find the right combination of words to search for. Suppose we have a free group on 2 generators $G = \langle a, ...
2
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0answers
45 views

Kernel of an homomorphism between two free groups

I am trying to prove that $G=\langle \alpha,\beta,\gamma \mid \alpha\beta\alpha^{-1}\beta^{-1}\gamma \rangle$ is isomorphic to $H=\langle \delta,\varepsilon \mid \hspace{0.5cm}\rangle$. Let N be the ...
2
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1answer
34 views

Suppose $G = G_1 * G_2$. let $c \in G$ and let $A = cG_1c^{-1}$. Show that $A\cap G_2 = \{1\}$

Suppose $G = G_1 * G_2$. let $c \in G$ and let $A = cG_1c^{-1}$. Show that $A\cap G_2 = \{1\}$. I think one must prove from contradiction, so suppose this intersection has another point $x \in G_2$ ...
2
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3answers
176 views

What is a free group element that is not primitive?

A primitive element of a free group is an element of some basis of the free group. I have seen some recent papers on algorithmic problems concerning primitive elements of free groups, for example, the ...
2
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0answers
33 views

Generate specific reduced words that “violate freeness”

Let $G$ be a group, and let $g_1,g_2\in G$ be nontrivial elements that do not commute. If $g$ and $h$ are not free as group elements, then the only a priori information this provides us is that there ...
2
votes
1answer
59 views

Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
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0answers
49 views

What is 'free algebra'?

I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring. But I still quite don't get it. Is a free algebra just a free group with additional operation ...
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20 views

Application of theorems 'about free groups'

What consequences have theorem Any nonzero subgroup of free group is free or some another similar theorems? P.S. especially not-group-theretic applications.
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3answers
77 views

Cayley graph is a tree iff group is free

I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S. For the direction '$\implies $' I see that they have assumed that there are two ...
3
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1answer
48 views

Rank of free group

Let $G$ be a free group with a basis $S$. Let $G'$ be the commutator subgroup of $G$. Define $S'=\{gG'\in G/G': g\in S\}$ Then, $S'$ is a basis for the free abelian group $G/G'$. Following the ...
1
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1answer
47 views

Definition of Polynomial ring through group rings

Is there any standardized/formal way that the polynomial rings $$R[x_1,x_2,\ldots]$$ $$R[x_1,\ldots,x_n]$$ $$R[x]$$ are defined. Would it be proper to say that $\text{PolyRing}(I)$, the polynomial ...
0
votes
1answer
46 views

A question about free product $\mathbb{Z_{2}}*\mathbb{Z_{2}}*\mathbb{Z_{2}}$

I want to use some examples to comprehension the definition of free product. Let $\mathbb{Z_{2}}$ be the integers $\{o, ...,m-1\}$ with addition modulo $m$ as the group operation and ...
0
votes
0answers
24 views

Subgroup of commutators of generators

My target is to prove that, in order to abelianize a group, it sufficies to take the subgroup generated by the commutators of the generators instead of the commutators of the whole group, to make the ...
0
votes
2answers
54 views

A problem on free products

If $G=A * B$ is the free product of two groups $A$ and $B$ and $g \in G-A$, then prove that $gAg^{-1} \cap A=1$. We know $A \cap B=1$, so if we write $g=a_1b_1a_2b_2 \ldots a_nb_n$, does not give me ...
0
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1answer
29 views

Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$

Prove that $PSL(2,\mathbb{Z})=C_2 \star C_3$. Now $C_2 \star C_3=\langle a,b\ |\ a^2, b^3 \rangle$ i.e. the free product. But how do I show that presentation of $PSL(2,\mathbb{Z})$ is this?
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0answers
36 views

the presentation of a group is well defined

In Hungerfold's algebra, Page 67, there is a definition of the presentation of a group using the language of free groups. Definition 9.4. Let X be a set and Y a set of (reduced) words on X. A group ...
2
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2answers
60 views

Number of normal subgroups from $F_{2}$ which factor groups are isomorphic to $D_{n}$

What is the number of normal subgroups of the free group $F_{2}$ whose factor groups are isomorphic to the dihedral group $D_{n}$?
2
votes
2answers
39 views

Kernel of epimorphism from $F_{k}$ to $Z_{n}^k$

How can be proved what all the epimorphisms from free group $F_{k}$ to $Z_{n}^k$ have the same kernel ? I guess where something must be done with the fact what $F_{k}$ is isomorphic to free product of ...
1
vote
1answer
49 views

What is a “right” automorphism?

Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads: If an automorphism of ...
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1answer
23 views

Is the Cayley graph of a free group dense on the Poincare disc?

In other words, every point inside the disc corresponds to a word (possibly of infinite length) of the free group; Is that correct? With this embedding:
2
votes
1answer
55 views

Prove that a group is isomorphic to the discrete Heisenberg group

Let $F_2$ be the free group of rank $2$ generated by $x$ and $y$. The discrete Heisenberg group $H$ is defined to be the factor group of $F_2$ modulo the relation that $[x,[x,y]]=[y,[x,y]]=1$. Let $G$ ...
2
votes
2answers
57 views

The free group given by $\langle a,b:a^2=b^3=e\rangle$ is not abelian.

Let $G=<a,b:a^2=b^3=e>$. I'm trying to show that $G$ is not abelian. $G$ is by definition given by $F(\{a,b\})/N$ where $F(\{a,b\})$ is the free group on 2 letters and $N$ is the smallest normal ...
1
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1answer
43 views

“Length” of an element in a free group

Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the ...
1
vote
1answer
58 views

Proof that free groups exist without the theory of finite words

I am interested in developing a formal proof that free groups exist, i.e. for any set $S$ there is a group $G\supseteq S$ such that for any group $H$ and any function $f:S\to H$ there is a ...
3
votes
1answer
128 views

Show that the group is trivial. [duplicate]

Show that the following group is identity: $$G=\langle x,y,z \mid xyx^{-1}=y^{2}\, , \, yzy^{-1}=z^{2}\, , \, zxz^{-1}=x^{2} \rangle.$$ This group is its own derived group. So all I get is group ...
11
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8answers
2k views

Why the group $\langle x,y\mid x^2=y^2\rangle $ 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.
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votes
1answer
36 views

T/F question on free groups

Is this statement True or False- If F is a free group with basis {$x,y$} and H is the subgroup generated by {$x^2,y^2,xy,yx$} then H is a free group of rank $3$. What should be my approach to solve ...
1
vote
1answer
32 views

How do I prove that a free group is generated by its base?

Let $G$ be a free group with a base $S$. How do I prove that $G=\langle S\rangle $ only using universal property? Here's how I proved this. Let $F(S)$ be the free group on $S$ constructed via ...