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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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
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0answers
19 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 ...
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2answers
39 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 ...
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1answer
11 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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26 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
52 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 $D_{n}$?
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2answers
28 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 ...
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1answer
38 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
14 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:
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1answer
38 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 $G$ ...
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52 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 ...
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1answer
35 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
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1answer
49 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
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1answer
109 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 ...
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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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1answer
29 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 ...
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1answer
27 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 ...
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0answers
33 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 ...
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2answers
38 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 ...
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2answers
38 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 ...
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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 $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
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1answer
53 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 ...
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82 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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37 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 ...
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1answer
67 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 ...
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3answers
57 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 ...
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0answers
51 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
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1answer
176 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
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0answers
42 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 ...
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1answer
41 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 ...
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2answers
82 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 ...
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0answers
33 views

Rreferences for free groups

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 ...
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79 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
2
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1answer
68 views

Generators of free groups

Ive been reading an introduction on free groups and have come across some difficulty if i have the free group $\mathbb{Z*Z}$ , then what would the generators of this group be? i am confused as ...
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1answer
62 views

Surjections from free groups

I am stuck on the following: How do I go about finding surjections from the free group of rank 2 $\mathbb{F}_2 = \mathbb{ Z∗Z}$, to the finite group of two elements $\mathbb{Z}_2$. Also, how would ...
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2answers
48 views

System of generators and surjective homomorphism

Let $G$ and $G'$ be groups and $\varphi:G\to G'$ a group homomorphism and $(g_s)_{s\in S}$ an indexed collection of elements of $G$ and is also system of generators of $G$. If $\varphi$ is surjective ...
2
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2answers
72 views

A simple question about free group

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. I have asked a question several days ago: Is $\mathbb{F}_{2}$ a subgroup of $\mathbb{F}_{3}$? ...
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2answers
22 views

Existence of Unique Homomorphism Implies Generating Set

The following question is taken from "An Invitation to General Algebra and Universal Constructions", p. 23 Ex. 2.1.2 (available online here). Let $G$ be a group and let $\{a,b,c\}\subseteq G$ such ...
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43 views

Prove that $A \times B \cong C * D$ is not possible

Prove that there is no group $G$ such that it has non-trivial expansion simultaneous in direct product and free product, i.e. $G = A \times B = C * D$. Check my attempt please. Let's take an ...
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1answer
37 views

Homomorphism between free group and symmetric group

Let $G:=S_5$ with elements $g_1=(12345)$ and $g_2=(12)$ and let $\varphi:F_2\to G$ be the unique homomorphism satisfying $\varphi(x_i)=g_i$ for $i=1,2$ Questions $\bullet$ Is ...
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1answer
31 views

Descending central series of free groups

What is the general term $\Gamma_k(F_n)/\Gamma_{k+1}(F_n)$, where $F_n$ is the free group of $n$ generators and $\Gamma_k(F_n)$ is the $k$th term in the descending central series of $F_n$?
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1answer
25 views

Finding free subgroups thanks to Lie algebras

Let $f : F \to G$ be a homomorphism from a free group $F$ to a group $G$. I heard that, in order to verify whether or not $f$ is one-to-one, it is possible to associate a Lie algebra $E_0^*(H)$ to any ...
2
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1answer
73 views

finite simple groups and free groups

It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? ...
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60 views

The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$

Show that: The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$. I've tried to apply the first isomorphism theorem to this by defining the ...
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32 views

center of free product of groups

Show the following: Let $G_1, G_2$ be groups and denote by $G_1*G_2$ their free product. Then center of $G_1*G_2$ contains only the neutral element. Have you some nice references about free ...
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1answer
40 views

Free Group Norms

Hello everyone, I'm trying to solve this problem, but I'm stuck... i don't quite understand the definition of the norm, If you guys can give me a better explanation, I would appreciate it, Thanks
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36 views

Count how many “free words” of a certain length reduce to the identity

Let $F_n$ be the free group with $n$ generators $g_1,\ldots,g_n$. I'm trying to settle the following: Question. For a fixed even integer $m$, is there a systematic way to count how many words ...
2
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2answers
45 views

Free Group generated by S is actually generated by S.

Consider the definition of free groups via the universal property: Definition. We say that the group $F$ is the free group generated by the set $S$ if there's a map $f:S\to F$ such that whenever ...
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45 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
4
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1answer
51 views

Which Words Are Part of a Free Basis of $F_n$?

Start with a free group on $n$ generators, $F=\langle a_1,\ldots, a_n\rangle$. If I write a word, $w$, in these generators, is there an efficient algorithm to determine whether or not $w$ can be ...