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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2
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1answer
62 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
35 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
votes
1answer
60 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
votes
1answer
53 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" means "are free generators of a free group". The ...
2
votes
2answers
58 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
35 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
32 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 ...
1
vote
2answers
48 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
votes
0answers
43 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
votes
1answer
33 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
152 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
votes
0answers
31 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
57 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 ...
0
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0answers
38 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 ...
1
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0answers
19 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
50 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
votes
1answer
43 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
vote
1answer
34 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
43 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
21 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 ...
1
vote
2answers
48 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
votes
1answer
20 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?
0
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0answers
30 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 ...
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2answers
59 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 $D_{n}$?
1
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2answers
31 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
44 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 ...
1
vote
1answer
15 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
44 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
56 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
38 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
52 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
117 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 ...
8
votes
8answers
1k views

Why is this group 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
33 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
30 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 ...
2
votes
0answers
35 views

Why $G$ is not free?

I have to show that $G=\langle x,y,z\ |xz=zx \rangle$ is not free. Now either I show it has an element of finite order which I dont see works here or I show it has no non-trivial defining relators ...
1
vote
2answers
44 views

Normalizer of an element of a free group

The problem says that Let F be the free group on $x_1,x_2, \ldots ,x_n$. Show that the normalizer/centralizer of any element $\neq 1$ is a cyclic group. The problem seems much harder to me as none of ...
2
votes
2answers
40 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
3
votes
0answers
14 views

Uniqueness of induced functions on reduced free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, that is, $F_n$ modulo the relation that $[x_i,x_i^g]=1$ for all $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
4
votes
1answer
55 views

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$. I've been trying to prove this given the definition of a free group $F$: given group $F$ and subset $X\subseteq F$, $F$ is free ...
6
votes
1answer
98 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.
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0answers
40 views

Two homomorphisms on the reduced free group

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, which is defined in the following two equivalent ways: $K_n=F_n/I$, where $I$ is the normal ...
5
votes
1answer
71 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 ...
1
vote
3answers
60 views

subgroup is saturated iff it is a direct summand

Let $A$ be a free abelian group of finite rank. Call a subgroup $B \le A$ saturated if for all $a \in A$ and positive integers $n$ such that $na \in B$, the element $a$ belongs to $B$. Call a subgroup ...
1
vote
0answers
54 views

Does a finite product of commutators of the form $[a,b]$ lie in $A∩B$?

1) If $G = A★B$ be the free product of two groups $A$ and $B$, then $A$ and $B$ need not be normal to $A★B$ necessarily, right? My Aim: I need to know if finite product of commutators of the form ...
3
votes
1answer
185 views

Free abelian group, matrix representation.

Let $n$ be a positive integer and let $A$ be a free abelian group and let $\{e_1,\dots,e_n\}$ be a basis of $A$. Let $B$ be a subgroup generated by $\{v_1,\dots,v_n\}$ and $M=(m_{ij})$ be a matrix ...
2
votes
0answers
44 views

Co-rank of a group with $a^2b^2c^2=1$

How do I calculate the co-rank of a group $$G=\langle a,b,c,\dots,z\mid a^2b^2c^2\dots z^2=1\rangle,$$ that is, finitely generated group with one relation? By co-rank, I mean the maximum rank of a ...
0
votes
1answer
44 views

Proving a homomorphism $\varphi: F \to G$ with F a free group and $\phi: G \to H$ a surjective homomorphism and $\psi: F \to H$ a homomorphism.

Let $\phi: G \to H$ be a surjective homomorphism and let $\psi: F \to H$ be a homomorphism with $F$ a free group with basis X. Prove that there exists a homomorphism $\varphi: F \to G$ such that $\phi ...
1
vote
2answers
100 views

An example of free group

Let $\alpha : \mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $\alpha(x)=x+2$ and $\beta:\mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $ \beta(x)=x/(2x+1)$. Show that the ...
1
vote
0answers
39 views

Rreferences for free groups [duplicate]

I have done free groups. I studied it from Rotman two semesters back. But this semester I am doing combinatorial group theory and obviously it starts with free groups. I have to revise Free groups but ...