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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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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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 ...
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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 ...
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1answer
26 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 ...
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39 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 ...
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20 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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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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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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27 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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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 $D_{n}$?
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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
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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
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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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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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53 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
36 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 ...
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1answer
51 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 ...
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111 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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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
29 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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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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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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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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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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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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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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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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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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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 ...
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1answer
177 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 ...
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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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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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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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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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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 ...
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1answer
70 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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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 ...
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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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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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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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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 ...
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1answer
75 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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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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34 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
41 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