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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Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
1
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0answers
19 views

Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
5
votes
1answer
42 views

How to Identify a Quotient of a Given Free Group

$\newcommand{\Z}{\mathbf Z}$ Problem. Let $G$ be the free group generated by three symbols $a, b$ and $c$, and let denote $G$ by writing $F(a, b, c)$. Let $N$ be the normal subgroup of $G$ ...
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0answers
15 views

All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
0
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0answers
23 views

Abelianizated free product of two groups

Given $$G=\mathbb{Z}_2*\mathbb{Z}_2=P(a,b\mid a^2,b^2)$$ among other things I wanted to show that this group is infinite, what I did is consider the words of the form $$abababa\ldots$$ they are all ...
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0answers
42 views

Index of free group of rank $k$ in free group of rank $n$

I thought about the following question What is the index of the free group of rank $k$, denoted $F_k$, in the free group of rank $n$, denoted $F_n$? Let's say for the moment, $k < n$, and ...
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0answers
21 views

Calculating fundamental groups using retracts

I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at ...
0
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0answers
7 views

Resources for free groups and surfaces

I'm looking for a good, clear, undergraduate level (if possible) reference (book, article, notes, etc) for the study of free groups, their topology and their relationship with surfaces. Thanks for ...
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1answer
23 views

Elements of tensor product

Let $R, S$ be rings such that $R$ is a subring of $S$ and $1_R = 1_S$. Let $N$ be a left $R$-module. Let the free $\mathbb{Z}$-module on $S \times N$ be $F_\mathbb{Z}(S \times N)$. Let $H$ be the ...
0
votes
1answer
34 views

Abelian Group as the quotient of a free Abelian Group

Is it true that every abelian group $G$ is the quotient of a free abelian group $F$? I think so, since every abelian group $G$ is the quotient of a free group $H$ under some relations, but some of ...
4
votes
1answer
30 views

Lifting an isomorphism of quotients to an automorphism of free group

Suppose that $F$ is a free group on $2$ generators and let $K,H \unlhd F$ be such that $F/H$ and $F/K$ are isomorphic. If $\phi \colon F/H \to F/K$ is an isomorphism, can it be lifted to an ...
3
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1answer
43 views

Frattini Subgroup of Free Group

It is perhaps around 1950, proved by Higman and Neumann that the Frattini subgroup of a Free group is trivial. I want to know, now, is there elementary proof of this? If yes, please write short ...
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27 views

detailed proof needed for |hom(F1, G)| = |hom(F2, G)| implies |X1| = |X2|

This is the real question. Let F1 be free on X1 and F2 be free on X2. Then F1 is isomorphic to F2 if and only iff |X1|=|X2|. (i.e. any two free groups are isomorphic.) This is the proof, i have so ...
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1answer
24 views

Prove that the presentation of a cyclic group of infinite order is $\langle a\mid \rangle$

Essentially, i need this for a third year mathematics project and originally i thought i just needed to have something like this: The group with this presentation is explicitly realized by the set of ...
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0answers
20 views

Free objects in the category of groups with operators.

Consider the category $\mathbf{C}$ of groups with a given operator domain $\Omega$. It's true that for every set $X$ there is a free object $G$ on $X$? I believe it's true and that the ...
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1answer
55 views

Use of ![1] syntax in GAP

What is the use of ![1] in GAP? It is used in many source files. ...
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2answers
82 views

Order of a generator element of Free Group in GAP

I am using the FreeGroup function in GAP ...
3
votes
1answer
56 views

What is the most general group possible?

I have read that free groups are the "most general" groups given some generators (from Wikipedia): The construction of a free product is similar in spirit to the construction of a free group (the ...
2
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1answer
36 views

Conjugacy of subgroups of free groups

Let $F(X)$ be the free group on a finite set $X$. Suppose that $H \subseteq F(X)$ is a subgroup where $[F(X):H]$ is finite. What can we say about the size of $[F(X):N(H)]$ where $N(H)$ is the ...
2
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3answers
57 views

Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
1
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1answer
48 views

Proof that $\mathbb{Z}^{\oplus \mathbb{N}}$ is isomorphic to its product with itself

I am trying to prove $\mathbb{Z}^{\oplus \mathbb{N}} \times \mathbb{Z}^{\oplus \mathbb{N}} \cong \mathbb{Z}^{\oplus \mathbb{N}}$ and will appreciate hints to approach the question. Background: ...
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1answer
43 views

Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
2
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0answers
50 views

Equation over free group

Let us consider free group $F(a,b)$ of rank two. I need to find a solution (or to prove that there is no one) over this group of the following equation: $$x^2[x^{2k},y]=a^2b^2,$$ where ...
3
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0answers
52 views

Basic properties of free product amalgamation of groups

I have some basic questions about properties of free product amalgamation of groups. Both can be phrased inside some fixed group $G$, which I am thinking of as having some very large infinite ...
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0answers
21 views

Inductive proof of associativity of free groups

I'm really struggling with the inductive proof of the associativity of free groups, given about halfway down page 6 of this pdf. The bit I'm not getting is this: Suppose now that bc involves a ...
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1answer
11 views

The order deduced from relations in $D_n$

If $D_n \triangleq \langle a,b | a^n=e, b^2=e, abab=e \rangle$, can it be proved that the order of $a, b$ is actually $n$ and $2$ respectively ? I mean can the relations on the right somehow after ...
3
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2answers
39 views

Showing a free abelian group is generated by its basis

I'm following Massey's "Basic Course in Algebraic Topology" and I'm stuck on his section explaining free abelian groups. He seems to be using a definition of free abelian groups that differs from most ...
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2answers
53 views

The rationals as an additive group is free? [closed]

Is there a set $X$ so that $(\mathbb{Q},+)$ are free on $X$ ?
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1answer
17 views

Condition under which one can extend a linearly independent set in a free $\mathbb{Z}$-module to a basis?

I'm having difficulty answering the following simple (?) question: Let $X = \mathbb{Z}^n$, and let $M$ be a submodule of $N$ closed under division - by which I mean that if $x \in X$ has $n x \in M$ ...
4
votes
2answers
83 views

There is a free group $F_2$ in $SO(3)$

I'd like an exposition of the proof that there is a subgroup of $SO(3)$ isomorphic to $F_2$, the free group on two elements. This is a key step in the proof of the Banach-Tarski paradox. The usual ...
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votes
1answer
20 views

Existence of unique homomorphism between free groups

We know that a free group F(S) on a set S comes equipped with a function $\alpha_S : S \to F(S)$. For any function $\beta: S \to G$, for some group $G$, there exists a unique homomorphism $\phi : F(S) ...
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1answer
117 views

What equational properties of a group only need to be checked on a generating set?

Let $G$ be a group and $S\subset G$ a generating set. Let $P$ (short for $P(x_1,\dots,x_n) = 1$) be an equational property that may or may not be satisfied by all $n$-tuples of elements of $G$. My ...
0
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1answer
28 views

How do I prove that a group with one generator and a single relation is isomorphic to $\mathbb{Z_m}$?

That is, if $G= \langle a \, | \, a^m = e \rangle$, then $G$ is isomorphic to $\mathbb{Z_m}$.
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1answer
40 views

What is the motivation behind free groups? [duplicate]

I have started studying free groups but cannot get motivation behind it. I know that a free group $F_A$ over the set $A$ consisting of all combinations or expressions that can be built from the ...
3
votes
1answer
30 views

Kernel of map equal to commutator, free group on two generators.

Let $G$ be the free group on two generators, and $K$ its commutator. We have that $G/K \cong \mathbb{Z} \oplus \mathbb{Z}$. This isomorphism is induced by $\theta: G \to \mathbb{Z} \oplus \mathbb{Z}$ ...
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0answers
23 views

A subgroup of free group

Let $F$ be free group on a finite set $S$. Let $H$ be the collection of words in $F$ which have even length when they are reduced to simplest form by cancellation. Then $H$ is a subgroup of index $2$ ...
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1answer
45 views

Cyclically reduced words in free groups

Let $F$ be a free group on $\{x_1,x_2,\cdots\}$, $w$ a word in $F$. Then $w$ is a finite expression in $x_i$'s and their inverses. By cancellation, $w$ can be reduced to simplest expression. Let ...
3
votes
1answer
72 views

Free group on 2 generators, how many subgroups of index $2$? [duplicate]

Let $F$ be a free group on two generators $x$, $y$. How many subgroups of $F$ have index $2$? What are the generators for each of these subgroups?
2
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1answer
20 views

Is the complete module product free? [duplicate]

Given a set $I$ and an $R$-module $M$, define the complete module product $M^I$ to be the set of all functions $I\to M$, with addition and scalar multiplication done componentwise. By contrast, given ...
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1answer
28 views

Showing that the free group (as a set of strings) is free

I'm working on a construction of the free group, and as the proof is taking me further afield than I'd expected I'd like some verification that I'm going the right direction and there is not some ...
1
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1answer
54 views

Free group and commutator question from Artin's book

Let $F$ be the free group on $x$, $y$ and $R$ be the smallest normal subgroup containing the commutator $xyx^{-1}y^{-1}$. a) Show that $x^2y^2x^{-2}y^{-2}$ is in $R$. b) Prove that $R$ is the ...
0
votes
0answers
26 views

Commuting elements in ${\rm PSL}(2,\mathbb{Z})$ are powers of some element

In free groups, it is easy to show that two commuting elements are powers of some element: if $x,y$ are in a free group, then $\langle x,y\rangle$ is free, being subgroup of a free group, with rank ...
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2answers
39 views

Writing specific word as product of commutators in free groups

Let $F$ be a free group on $\{a,b,c,\cdots\}$. Then a word $$a^mb^nc^k\cdots \hskip1cm \mbox{(finite expression)}$$ lies in commutator subgroup $[F,F]$ if and only if sum of powers of $a$ is zero, ...
0
votes
1answer
48 views

Extension of a Set to Basis of Free Group

Let $F$ be free group on $\{x,y\}$. Consider the element $x^2$. Does there exists $z\in F$ such that $\{z,x^2\}$ generate whole $F$? I thought in following direction. $x^2$ is not in commutator ...
3
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1answer
37 views

Free Generators (Basis) for a Normal Subgroup of a Free Group

Let $F=\langle a,b\rangle$ be a free group of rank $2$. Its commutator subgroup has a nice free-basis: $$[a^m,b^n], \,\,\,\,m,n\in\mathbb{Z}.$$ Instead of $[F,F]$, we consider another simplest normal ...
2
votes
2answers
70 views

On powers in free groups

Let $G$ be a free group and $a,b\in G$ be such that $a^n=b^n$ for some $n\in\mathbb{Z}$. Then $a=b$. I followed following way: if rank(G) is 1, then $G$ will be cyclic, and assertion is true. ...
0
votes
1answer
39 views

Commutator Subgroup of Free Group

In the Combinatorial Group Theor-Lyndon, Schup, the authors say (start with) Informally, a group is free on a set of generators if no relation holds among these generators except trivial relations ...
2
votes
1answer
38 views

Discrete subgroups of $\mathbb R^n$ are free

Let $G$ be a nonzero subgroup of the additive group $\mathbb{R}^n$. Assume $G$ is discrete in the sense that for any $x \in G$, there exists an open set $U \subset \mathbb{R}^n$ such that $U \cap G = ...
2
votes
1answer
27 views

Problem involving Generators and Relations from Michael Artin's book.

The question is: "Can every finite group G be presented by a finite set of generators and a finite set of relations?" I think the answer is yes, but I can't find a general way to decompose every ...
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1answer
27 views

A word problem in abelian $p$-groups

Let $G=\langle a,b,c\rangle$ an abelian $p$-group of rank $3$ with only following more relations between $a,b,c$: $$a^{p^2}=b^{p^2}=c^{p^2}\neq 1, \mbox{ and } a^{p^3}=1.$$ Then, consider the ...