Tagged Questions

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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Normal closure of a subgroup of a free group.

Let $G$ be a finitely generated free (nonabelian) group, $H$ a subgroup generated by some of the generators of $G$, and $a: G\to AG$ be the projection to the abelianization $AG:=G/[G,G]$. Is it true ...
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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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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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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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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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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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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 $F_a$ ...
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 ...
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 ...