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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Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
4
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0answers
70 views

How to get a Presentation of a Group

$\newcommand{\R}{\mathbf R}$ Let $G$ be the group of homeomorphisms of $\R^2$ generated by $g$ and $h$, where $g(x, y)=(x+1, y)$ and $h(x, y)=(-x, y+1)$. To show that $G\cong \langle a, b|\ ...
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2answers
29 views

Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we ...
0
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1answer
37 views

Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted ...
0
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3answers
34 views

What is the logic in Free groups here

Free groups have been hurting my brain for months and this one in particular, I cannot comprehend what is going on with logic $H= \langle a,b\mid a^{-1}ba=b^2,b^{-1}ab=a^2\rangle$. Show that $H$ ...
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3answers
87 views

An exercise to understand free group

I am new to the concept of free groups, reading Artin's Algebra but completely lost. So I hope I can learn from concrete examples instead of theorems corollaries. So I jumped to the exercise, and here ...
1
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1answer
106 views

GAP and Free group

Let $H$ and $K$ be finitely generated subgroups of the free group $F(n)$ such that $H\subseteq K$. We Know that $K$ is a free group. Now we choose basis for $K$ and $H$ and I want to rewrite the ...
5
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1answer
42 views

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$. The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that ...
9
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2answers
113 views

Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
0
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1answer
41 views

Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
5
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1answer
36 views

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 ...
0
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1answer
29 views

A monoid is universal (or free) over its generators iff no nontrivial relations hold among its generators

Let $X$ be any set. A monoid $M$ is called universal over $X$ iff $X \subseteq M$ and for every other monoid $N$ and function $\varphi : X \to N$ there exists a unique extension $\varphi : M \to N$ of ...
1
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1answer
28 views

An odd map having odd degree little issue with the proof

In the proof below of hatcher I agree that $Im(\tau) \subset Ker(p_{\#})$. However, what I don't understand why is the other inclusion true ?
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1answer
27 views

Exact sequence of free abelian groups, $\sum_{i=0}^n(-1)^i\text{rank}(F_i)=0$.

This question is from Rotman's Introduction to the Theory of Groups: (i) Suppose we have an exact sequence of free abelian groups $A\to B\to C\to D$ with maps $f,g,h$ in between. Show $B\cong ...
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10answers
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I don't understand what a “free group” is!

My lecture note glosses over it really, introduces it and says "well it intuitively makes sense" but I say, nope it doesn't. Free groups on generators $x_1,...,x_m,x_1^{-1},...,x_m^{-1}$ is a ...
1
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1answer
35 views

Abelianization of free product is the direct sum of abelianizations

I define $\text{Ab}(G)=G/[G,G]$ where $[G,G]$ is the commutator subgroup. I want to show that $$\text{Ab}(G_1*G_2)\cong \text{Ab}(G_1)\oplus\text{Ab}(G_2)$$ This page gives a categorical proof, but I ...
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3answers
40 views

Free group is equal to $\mathbf{Z}^2$

I would like to show that the free abelian group $\langle a, b : aba^{-1}b^{-1} = 1 \rangle$ is equal to $\mathbf{Z}^2$. A friend of mine suggested to use the Van-Kampen-theorem. However, I was hoping ...
2
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2answers
59 views

$G=\langle a,b \mid abab^{-1}\rangle$ and $H=\langle c,d \mid c^2d^2\rangle$ are isomorphic (Can't use Seifert/van Kampen Theorem)

I'm reading up on Algebraic Topology in preparation for a summer course, and learning about the classification of surfaces I ran across this problem: Show that the groups $G=\langle a,b \mid ...
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4answers
50 views

Homomorphic image of a free group

Given a group $G$ with a generating set $S$, such that $|S|=n$, I need to prove that $G$ is a homomorphic image of $F_n$. Right now I'm just looking for any tips for how to even start this proof or ...
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2answers
70 views

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 $\varphi(x) \neq e$ ...
2
votes
1answer
21 views

Showing that $(\phi^2(\vec x),\phi(\vec x),\vec x)$ is a free family, $\phi^2(\vec x)\neq \vec 0, \phi^3(\vec x)= \vec 0$

Let be $A\in M_4(\mathbb C)$ such that with $O_4$ representing the null matrix of rank $4$, $$A^2\neq O_4 \mbox{ and } A^3=O_4$$ Let be $\phi$ the endomorphism of $\mathbb C^4$ ...
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0answers
31 views

Subgroup of free group who's conjugacy classes are disjoint from the generators

Let $F(S)$ be the free group generated by the set $S$. Suppose $H \subseteq F(S)$ be such that $H^{F(S)} \cap S = \emptyset$. That is, for any generator $s \in S$ we have that $s \notin gHg^{-1}$ ...
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1answer
36 views

A question of quasi-isometric of free groups [closed]

For two free groups with finite ranks, are they quasi-isometric to each other?
2
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0answers
17 views

Free groups and homomorphisms [duplicate]

I'm currently stuck on this problem: Let $m$ and $n$ be positive integers, and $F_k$ be the free group generated by $k$ elements. Prove that there is a homomorphism from $F_n$ onto $F_m$ if and ...
2
votes
1answer
50 views

What is the presentation of $\mathbb{Z}_n$

I am given $\langle a,b | ab=ba, b^6=1 \rangle$ and I am supposed to compute the group that has this presentation. After racking my brains for a long time, the only thing I can come up with is ...
4
votes
1answer
34 views

Relation between lower descending $\gamma_5(F)$ and derived $F"$ in a free group

Let $F$ be a free group (I'm interested in the case $F$-finitely generated). Let $\gamma_n(F)$ be the corresponding terms of the lower descending series, so for example $\gamma_4(F)=[F,[F,[F,F]]]$. ...
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0answers
21 views

Homomorphisms between two universal free groups [duplicate]

I'm trying to prove that, for positive integers, $m$ and $n$, there is a homomorphism from $F_n$ onto $F_m$ if and only if $m$ is less than or equal to $n$ (where $F_n$ is the universal free group ...
4
votes
1answer
31 views

Any countable free group is embeddable into a free group of rank $2$

I have found this proof of the fact that any countable free group is embedable in a free group of rank $2$ (see the last page, Proposition 2). But isn't this proof incorrect? First off, it says ...
2
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3answers
66 views

Homomorphism between free groups

Let $F_a$ be the group which is freely generated by $a$ elements. How to show that there is a homomorphism from $F_a$ onto $F_b$ if and only if $b\le a$? I was thinking one possibility is if $F_a$ ...
3
votes
0answers
39 views

Let $G= G_1 * G_2$, where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$. Then $m$ and $n$ are uniquely determined by $G$.

I'm having trouble understanding the following problem from Munkres' Topology. I have shown (a) and (b) below, for (b) I got $k=\max(m,n)$, but I don't know what I need to prove for (c). In fact, what ...
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1answer
47 views

Free group presentation

So a free group is just a set $G$ with relations corresponding to the group axioms. Normally these relations are omitted from the presentation of a free group. The identity and inverse are easy but ...
0
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2answers
83 views

Definitions of free monoids and free groups

Recall that a word over the alphabet $A\cup A^{-1}$ is a sequence of elements of $A\cup A^{-1}$ where $A^{-1}=\{x^{-1}:x\in A\}$. The set of all the words over the alphabet $A\cup A^{-1}$ equipped ...
2
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1answer
32 views

Equivalence between two comeager sets related to free groups

Let $G$ be a non discrete Polish group. For every $n\ge 2$ equip $G^n$ with the product topology. Saying that $F_n=\{(g_1,\dots,g_n)\in G^n: \{g_1,...,g_n\}$ freely generates a free subgroup of rank ...
3
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1answer
53 views

If I say $\{g_1,\dots,g_n\}$ freely generates a subgroup of $G$, does that mean the elements $g_i$ are all distinct?

Let $G$ be a group and let $g_1,\dots,g_n$ elements of $G$. If I say that $\{g_1,\dots,g_n\}$ freely generates a free subgroup of $G$, say $H$, do I mean that the rank of $H=n$ or should I consider ...
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1answer
62 views

Free product of the powerset algebras of group orbits. Interpretation

I am trying to interpretate the following sentence in the context of measures on groups and algebras, from J. Pawlikowski on "The Hahn-Banach theorem implies the Banach-Tarski paradox" Let $F$ be ...
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2answers
67 views

The graph of free product group. [closed]

For the free product $A*B$=G, where $A$ and $B$ are groups, there is a graph defined by: the edge set E(G)$\backsimeq$G and the vertex set V(G)$\backsimeq$G $G/A \bigsqcup G/B$, and to ...
3
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1answer
50 views

The conjugacy problem of finitely generated free group

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?
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3answers
150 views

Precise definition of free group

I have seen the definition of a free group go like this: Let $S = \{s_i : i\in \mathbb{N} \}$ be a countable set. Let $S^{-1}$ be the set $\{s_i^{-1}: i\in \mathbb{N}\}$. Here one is to understand ...
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0answers
26 views

Free group on 3 generators is a subgroup of free group on 2 generators [duplicate]

I think this true but can it be proven in an explicit way? A group, G, is called Free if there is a subset S of G such that any element of G can be written in one and only one way as a product of ...
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1answer
52 views

What is the difference between free groups and a free product?

I am encountering free products for the first time in Algebraic Topology during the discussion of van Kampen's theorem, and I can't seem to tell the difference between a free product of groups, and a ...
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0answers
36 views

Rank of Non-Abelian Quotient Group

Let $G$, $G'$ be two (non-Abelian) finitely generated free groups, and $H$, $H'$ be their finitely generated normal subgroups respectively. I want to know if the following statement is true: $G/H ...
4
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0answers
43 views

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 ...
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1answer
35 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
48 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
22 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 ...
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0answers
26 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
49 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
22 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 ...
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1answer
37 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
40 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 ...