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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About the generators of a free group

Suppose we know that $G$ is a free group of rank $n$ and that $\{g_1,...,g_n\}$, with all the $g_i$ distinct, is a system of generators for $G$. Are we sure that it is also a $\textbf{free}$ system of ...
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Cyclically reduced words

This is just a reference request. I'm trying to find out whether there are some well developed notes/theory out there (books and the like) focusing on cyclically reduced words in groups. Quickly ...
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About an article regarding free groups

I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let $G$ be a non ...
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Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
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To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$ \bigcap_{N \in \mathcal{N}} N \ = \ \{ e\} $$ I know how free ...
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Problem in understanding some steps in proof

The problem here is about categorical construction of free groups, as in Lang's algebra (p.66-68). Theorem: For any set $S$, there exists free group $(F,f)$ determined by $S$ (here $f:S\rightarrow F$)...
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Formal construction of free groups and objections in arguments

For simplicity, consider $X=\{a,b\}$. Let $Y$ be another set in bijection with $X$, and write its elements to be $a^{-1},b^{-1}$. Let $W(X)$ be the collection of all words in $a,b,a^{-1},b^{-1}$, ...
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Are Free Groups the “Smallest Group” Containing their Generators

I apologize if this is a duplicate; I was not sure how to search for this. When I say "the smallest group" I mean unique up to isomorphism of course. Specifically, is "the smallest group containing ...
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On a property of generating set for groups

Consider the following paragraph from Lang's Algebra: My question is about converse of one statement, which I was not able to prove. Question: If $f(S)$ do not generates $F$, then can we find ...
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What is significance of this proof of existence of free groups (Lang's Algebra)

There are different proofs of existence of free groups. While reading Lang's Algebra, it caught my attention towards proof of this theorem by first bracket statement in proof: Later I went on ...
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About Structure of Free Algebra over $K$

In MIT Course No. $18.712$, Associative Algebra $A$ is defined as a vector space over a field $K$ with a bilinear associative map $A \times A \to A$, $(a,b) \to ab$. Then some examples are given, ...
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solve a specific word problem in free groups

Let $F_2=\langle a, b\rangle$ be the non-abelian free group with two generators and $e$ is the neutral element in $F_2$. Given $g\in F_2, k\geq 2$ an integer. I want to know how to solve the word ...
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A proposition about free product

My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses. $\textbf {...
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On the definition of free products

I am a little confused about the definition of free products. Given a collection of groups $\{G_\alpha\}_\alpha$ in order to create their free product, I don't understand what properties these $G_\...
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About rank of free module compared to spanning and LI set

Let M be a free R-module where R is commutative ring with unity. Q1: If M is FG by n vectors. Will rankM $\leq$ n? Q2: If M has a LI set $S$. Will #$S \leq$ rankM? What if R is PID? Are above ...
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Automorphisms of Free Groups of Finite Order

The automorphism group of a free group (of finite rank) is known (see this). The group is infinite, if the number of generators is at least $2$, since there are automorphisms of the form $x_i\mapsto ...
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To prove that a hopfian group is not free.

In the book "presentation of groups by johnson" in page $36$ they are trying to obtain a group that is locally free but not free (Example $2$).They have proved that the group $U=\bigcup_{n \in \mathbb{...
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If a subset of a free group $F$ is Nielsen reduced, then it is a basis of $F$. Is the converse statement true? [closed]

If a subset of a free group $F$ is Nielsen reduced, then it is a basis of $F$. Is the converse statement true? I mean if I take a basis $U$ of $F$, then is it true that it has to be Nielsen reduced? ...
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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. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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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|\ b^{-1}...
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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 ...
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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 ...
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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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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 ...
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117 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 ...
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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 ...
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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 ...
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46 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 ...
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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 $...
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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 ...
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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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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 \text{...
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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 ...
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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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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 ...
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$G=\langle a,b \mid abab^{-1}\rangle$ 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 abab^{-...
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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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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 $\varphi(x) \neq e$ ...
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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 $\mathbb C^4$ canonically ...
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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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A question of quasi-isometric of free groups [closed]

For two free groups with finite ranks, are they quasi-isometric to each other?
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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 ...
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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 $\...
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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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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 ...
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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 $w=b^{...
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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 $F_a$ ...
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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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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 ...
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Definitions of free monoids and free groups [closed]

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 ...