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Questions tagged [free-groups]

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 $F_n$.

Free groups are the free objects in the category of groups. This means that if $S$ is some set such that there exists a function $f: S\rightarrow G$ where $G$ is some group then there exists some group homomorphism $\varphi: F_S\rightarrow G$ such that the following diagram commutes,

The universal property of free groups - from Wikipedia

The universality of free groups implies the set $S$ which they are generated by is important, and indeed one can view a free group over the set $S$ as the set of all words over $S^{\pm 1}$ under the operation of concatenation. This leads to the theory of group presentations.

Free groups can be classified up to isomorphism by their rank. Thus, we can talk about the free group of rank $n$, denoted $F_n$.

The standard (classical) reference for free groups is the book "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations" by Wilhelm Magnus, Abraham Karrass and Donald Solitar.

Note: diagram taken from Wikipedia.

942 questions
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A Quotient of Free Group

If $F$ is a free group on a finite set $S$, then the squares in $F$ generate a normal subgroup $N$ and $F/N$ is elementary abelian $2$-group of order $2^{|S|}$. Let $F$ be free group on infinite set $S$, and $N$ the normal subgroup generated by…
p Groups
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Proving that a group generated by x,y and z and a given relation is actually free

I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group. What are some usual ways of going about this kind of problem? Just hints please! Regards
user50948
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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 $$ \sqrt{(p^2+q^2)(r^2+s^2)}=\sqrt{(pr+qs)^2+(ps-qr)^2}. $$ Is $K$ with…
Chung. J
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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. Hence, there exists a bijection $f:X \to X'$. Let $A…
Gustavo
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Uniqueness of reduced form

Let $X$ be a non-empty subset and $F(X)$ the free group on $X$. If $z\in F(X)$, it can be expressed in the form $z=x_1^{\epsilon _1} \cdots x_n^{\epsilon _n} $ where $\epsilon _i =\pm 1$, $x_i\in X$ and it contains no pair of consecutive symbols of…
user73564
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Finding a quotient of the free group

Consider the free group $F_S$ over a set $S$. Let $x \neq e$ be an element of $F_S$. Is it true that there is a group morphism $\varphi : F_S \to G$ to a finite group $G$ such that $\varphi(x) \neq e$ ? What happens if, instead of a single element…
vizietto
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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?
Yui
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The quotient of a free group, of rank $n$, and its commutator subgroup is isomorphic to $\mathbb{Z}^n$

Let $F$ be a free group of rank $n$. Let $G$ be the commutator subgroup of $F$. I need prove that $F/G\cong\mathbb{Z}^n$. I have tried with the isomorphism theorem: With $\varphi$, I send the $x_i$ element in the basis of $F$ to the vector with $1$…
user95747
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Construction of a free group

I am studying Algebraic Topology, and I need to understand the concept of 'free groups'. I've read the definition ,a free group is a collection of words formed from a set, called a generating set and it satisfies all of the group axioms. The problem…
しずか
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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 my attempts are anywhere working. Can somebody…
Bhaskar Vashishth
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Free groups and injective function

We have the following about free groups: Given a non-empty set $X$, it is possibly to find a group $G$ and a function $\sigma :X\to G$ such that $\sigma (X)$ algebraically generates $G$ and for every group $H$ and every function $f:X\to H$ there…
user73564
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Why are these two free groups isomorphic?

Let $p$ be a fixed prime and $\Gamma _2(p)$ the multiplicative groups of all matrices $ \begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ where $a,b,c,d$ are integers, $a,d$ are equal to 1 module p and $b,c$ are multiples of $p$. We consider $u =…
user73564
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Presentation of free group

I want to prove that $$\cong \mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$$ I already show $a^2=1$, but I don't make sure that $a\neq1$. How can I prove this? Any help would appreciated. Thanks.
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What is $[[H, H], [H, H]]$, where $H$ is the free group on two elements $a,b$?

I think that when $H$ is the free group on two elements, $[[H, H], [H, H]]$ is the trivial group. I feel that this shouldn't be too hard to prove/disprove, but I've got nothin'. Ideas?
rjkaplan
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Prove a given group is free

From Hungerford's "Algebra": What type of tools does one have to tackle a problem like this? I seem at a loss at how to show a group is free at all. One can consider each group as the homomorphic image of a free group, but how does one work in the…
asd
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