# 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 $F_n$.

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### The free generating sets of two isomorphic free groups must have the same cardinality [closed]

Claim: If $F(X_1) \cong F(X_2)$, then $|X_1| = |X_2|$. Proof: Look at the sets $\text{Hom}(F(X_1), F_2)$ and $\text{Hom}(F(X_2), F_2)$ of group homomorphisms to the field $F_2$ with $2$ elements. ...
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### Remark after proof of existence of free groups (page-21,Clara loeh)

Context Definition 2.2.4 (Free groups, universal property). Let $S$ be a set. A group $F$ containing $S$ is freely generated by $S$ if $F$ has the following universal property: For every group $G$ ...
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### Details in existence of free groups proof (Clara Loeh,pg-22,23)

I have a few doubts in the proof of existence of free groups given by Clara Loeh in page-22,23 of her Geometric Group theory book. Theorem 2.2.7: Let S be a set. Then there exists a group freely ...
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### Can a space with $\pi_1(X)=F_2$ have a nontrivial covering space which is homeomorphic to $X$?

Any finite sheeted connected cover of the circle is again homeomorphic to a circle. On the group level, this is consistent with the fact that subgroups of $\mathbb Z$ are isomorphic to $\mathbb Z$. ...
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### What does the author want to say about generators and relations in group theory? ("Algebra 1st Edition" by Michael Artin) [closed]

I am reading "Algebra 1st Edition" by Michael Artin. I want to know about generators and relations because I think I need to know about generators and relations when I use the GAP software. ...
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### Ping-pong lemma assumptions

The ping-pong-lemma for subroups is often stated with the following assumptions, e.g. taking this one from Wikipedia: Let G be a group acting on a set $X$ and let $H_1, H_2, ..., H_k$ be subgroups of ...
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### If $X$ is an infinite set, then $|X|=|F(X)|$, where $F(X)$ is the free group on $X$

I am struggling to prove that for an infinite set $|X|=|F(X)|$, where $F(X)$ is the free group on $X$. I can see the obvious injection $X \to F(X)$ but struggling to see the converse. Any help would ...
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### Free left $R$-module can be endowed with $(R,R)$-bimodule structure. Basis-dependent definition?

Let $R$ be a non-commutative ring. If we have a free left $R$-module $F$, we can consider a basis $\{e_i\}_{i\in I}$ in $F$ and endow $F$ with a $(R,R)$-bimodule structure in the following way: For a ...
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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 ...
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### A group is locally free exactly when its finitely generated subgroups are free

This is Exercise 6.1.9 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search for "locally free finitely generated" in the group theory tag, ...
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### Order of a quotient of a free abelian group

Let $G\subseteq \mathbb{C}$ be a free abelian group of rank $n$ and let $p$ be a prime. Then, we know that $$|G/pG|=p^n$$ and in fact for this we don't even need $p$ to be a prime. Suppose now that ...
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### On the meaning of "$F^2$" in Robinson's Exercise 6.1.7.

I am confused about some notation in Exercise 6.1.7 of Robinson's, "A Course in the Theory of Groups (Second Edition)". Here is the exercise: Exercise 6.1.7: Let $F$ be a free group of ...
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### Find rank of subgroup of $F_2 = \langle x,y\rangle$ generated by $xyx^{-1}y, yxy^{-1}x$.

This is from an exercise in a group theory course I'm taking, I believe the subgroup of $F_2 = \langle x,y\rangle$ generated by $a = xyx^{-1}y$ and $b = yxy^{-1}x$ is isomorphic to $F_2$. What I've ...
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### Describe all path-connected $3$ fold covers of $X=S^1 \vee \mathbb{R}P^2$

Describe all path-connected $3$ fold covers of $X=S^1 \vee \Bbb{R}P^2$ (please justify why your list is exhaustive). Which are regular (i.e. normal) and why? I get that $X$ has a universal cover, ...
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### Rotman's Algebraic Topology Lemma 9.11

This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology". The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are ...
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### Problem handling free groups in algebraic topology

Trying to compute the fundamental group of a topological space $X$ I have come to the equality $$\pi_{1}(X)\cong\frac{\ast_{i=1}^{n}\mathbb{Z}}{G}$$ where $\ast$ means taking the free product ($n$ ...
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### Every element in a free group is conjugate to a cyclically reduced word

Given a free group $F$ generated by a set $X$, which by definition is the set of reduced words in $X \cup X^{-1}$, with reduced concatenation of words, I've come across a statement that says every ...
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### Torus is the only closed orientable surface regularly covered by punctured plane

Let $\Sigma_g$ be the closed orientable surface of genus $g$. There is no covering map $p\colon\Bbb R^2\backslash \mathbf 0\to \Sigma_g$ so that $p_*\pi_1(\Bbb R^2\backslash \mathbf 0)$ is a normal ...
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### Center of a quotient of a free group

I encountered the following statement while going through an old result by C.F.Miller (Ref. The Schur Multiplier, Gregory Karpilovsky, Theorem 2.6.6, Page 72): Let $G=F/R$ be a presentation of an ...
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### Lee's proof of the rank theorem for abelian groups

I am going through Prof. Lee's "Introduction to Topological Manifolds" second time through, trying to do all the exercises and problems. My question is about a proof of the rank theorem for ...
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### In $F=F^{ab}(A)$, define $f\sim f'$ if and only if $f-f'=2g$ for some $g\in F$. Show $F/\sim$ is finite if and only if $A$ is finite.

this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following: Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence ...
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### Normal covering spaces of the wedge sum of $n$ circles

Exercise 1.31 in Hatcher's Algebraic Topology states the following: Show that the normal covering spaces of $S^1 \vee S^1$ are precisely the graphs that are Cayley graphs of groups with two generators....
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### If $a^n=b^n$ with $n>1$ in a free group, then $a=b.$
Edit : the free group 𝐹𝑆 consists of all reduced words that can be built from members of 𝑆 and formal inverses of members of 𝑆. Prove: If $a^n=b^n$ with $n>1$ in a free group, then $a=b.$ My ...