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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Showing a certain generator of a free group is primitive

Let $F$ be a free group generated by a subset $\{a_0, \ldots, a_n\}$ (not necessarily freely). Suppose the map $\phi \colon F \to \mathbb{Z}$ given by $\phi(a_0) = 1$ and $\phi(a_i) = 0$ for all $i &...
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If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.

If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one. Definition. A path-connected space whose fundamental group is isomorphic to a given group $G$ and which has ...
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Set of homomorphisms on a free abelian group is a free abelian group.

If $G$ is a free abelian group with rank $n$, I need to show that ${\rm Hom}(G,\mathbb{Z})$, set of all homomorphisms is also free abelian group of rank $n$, My work: Since $G$ is free abelian group ...
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Existence of automorphism taking a word to its inverse

Let $F_2$ be free group on free generators $\{x,y\}$. We know that the inverse of $[x,y] = xyx^{-1}y^{-1}$ is $[y,x]$ and if we take automorphism $\phi$ generated by $\phi(x)=y, \phi(y)=x$ then $\phi$ ...
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Diophantine equations and equations over free groups

I'm interested in quadratic diophantine equations such as the ones addressed [here] 1. I have come through a series of papers that solve equations over groups. For instance, here. Since $(\mathbb{Z},+)...
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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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1 answer
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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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2 answers
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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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Finding relators of a matrix group

Let $f_1,\dots,f_n$ be maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $f_i(x) := a_ix + b_i$ with $a_i,b_i \in \mathbb{Q}$. We construct the transformation group $G = \langle f_1, \dots, f_n \...
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Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$

$\newcommand{\Z}{\mathbb Z}$ Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
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What is the most concise way to present a particular subgroup of $F_2$?

Let $A\leq F_2$ where $F_2$ is the free group on $\{a,b\}$. Assume $A$ is generated by words on $a,b$ such that the total powers of $a$ are $3x$ for some $x\in \mathbb{Z}$ and the total powers of $b$ ...
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Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group?

Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group? Let's say that for two groups $A$ and $B$ we have that $A\times B \cong F_n$ where $F_n$ is the free group ...
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3 answers
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Is there a such thing as a free group generated by a free group?

Is there a such thing as a free group generated by a free group? Let $F(A)$ be a free group generated by the elements of a set $A$. If we momentarily consider $F(A)$ to be simply a set of words on ...
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Is there any proof of $\#(F/N)=2n$ which doesn't use any group other than $F/N$ itself? (Michael Artin "Algebra 1st Edition")

I am reading "Algebra 1st Edition" by Michael Artin. The following proposition is Proposition (8.3) on p.221 in this book. (8.3) Proposition. The elements $x^n,y^2,xyxy$ form a set of ...
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Question about a proposition about free groups, generators and relations. Is it true or false that $N=\ker\phi$ holds? Michael Artin "Algebra 1st Ed."

I am reading "Algebra 1st Edition" by Michael Artin. I feel free groups, generators and relations are very difficult. The following proposition is Proposition (8.3) on p.221 in this book. (...
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subgroup generated by n-th powers of elements in a free group [duplicate]

Let $F=\langle x_1,\ldots,x_n \rangle $ be a finitely-generated free group. For $n \in \mathbb{N} $, let $H_n$ be the subgroup of $F$ defined by: $H_n= \langle x^n|x\in F \rangle$. Is it true that $|F:...
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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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Any injective homomorphism $F_n\to F_n$ with image of finite index is bijective ($n\geq 2$)

$\DeclareMathOperator{\im}{Im}$ Let $\Phi: F_n\to F_n$ be an injective homomorphism between free groups with $n\geq 2$ and $\im{\Phi}$ having finite index. Prove that $\Phi$ is bijective, i.e. $\im{\...
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Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$

Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$. This is a follow-up to my post from yesterday regarding the kernel K of a map $\Phi: ...
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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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3 votes
2 answers
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What is the kernel of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$?

What is the kernel $K\leq F_2 = \langle a,b \rangle$ of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$ given by $a \mapsto (1+2\mathbb{Z},0+3\mathbb{Z})$ and $b\mapsto (0+2\mathbb{Z}, 1+3\...
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5 votes
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Growth of balls vs growth of spheres in hyperbolic groups

Let $G$ be a finitely-generated group equipped with a word-metric. Let $B_n$ and $S_n$ be the $n^{\mathrm{th}}$-ball and $n^{\mathrm{th}}$-sphere, respectively, with respect to the given metric. ...
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Is $\Bbb Z^n$ isomorphic to $\Bbb Z*\Bbb Z*\Bbb Z ...$, $n$ times?

Let $\mathbb{Z}^n$ be the group formed by the external direct product of $\mathbb{Z}$ taken $n$ times, and let $A_n$ be the group formed by taking the free product of $\mathbb{Z}$ $n$ times. Then, is $...
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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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2 votes
2 answers
94 views

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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Question about f.g. normal subgroups of f.g. free groups

Suppose $H$ is a finitely generated normal subgroup of a finitely generated free group $F(X)$ with basis $X = \{ x_1 , \dots , x_n \}$. Fix $1 \leq i \leq n$. Is it true that the set of right cosets $\...
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3 votes
2 answers
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A certain free product of groups is virtually torsion-free

Suppose that $G_1,\ldots,G_n$ are finite groups, and $m\geqslant 0$ is some integer. Set $$G=G_1\ast\cdots\ast G_n*F_m,$$ (where $F_m$ is the free group on $m$ generators). Then, is $G$ virtually ...
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Show that the kernel of a homomorphism from $G_1*G_2$ to $G_1\times G_2$ is free on $S$

Show that the kernel of $\omega: G_1*G_2 \to G_1 \times G_2$ given by $\omega(g_1g_2)= \iota_1(g_1) \cdot \iota_2(g_2) = (g_1,g_2)$ is free on $S=\{ gg'g^{-1}g'^{-1}, 1\neq g \in q_1(G_1), 1 \neq g'\...
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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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1 vote
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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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2 votes
1 answer
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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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1 vote
1 answer
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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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4 votes
1 answer
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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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4 votes
1 answer
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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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1 vote
1 answer
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Seifert-Van Kampen $S^1 \vee S^1$

I would like to use the fact that if two path-connected pointed topological spaces $(X,p)$ and $(Y,q)$ admit two contractible open neighbourhoods of $p$ and $q$, then $$ \pi_1(X\vee Y) = \pi_1(X)*\...
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-1 votes
2 answers
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Is $\mathbb{Z_5}$a free abelian group ? Yes/No [closed]

Is $\mathbb{Z_5}$ a free abelian group ? My attempt: I think $\mathbb{Z_5}$ is free abelian group By the definition of free abelian group $X$ generates $G$, and $n_1x_1 +n_2x_2 +\dots+n_rx_r=0$ ...
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9 votes
1 answer
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Prefixes of a word multiplying to the identity in a free group

Let $A$ be a finite alphabet, and let $w \in (A \cup A^{-1})^\ast$ be a freely reduced word over the alphabet $A$ and formal inverse symbols $A^{-1}$. Suppose $w$ is non-empty. Can there ever be non-...
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3 votes
1 answer
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Normal closure in free group

Let $G=\pi_1(\mathbb{C}-\{z_1,z_2\})$ be the fundamental group of the plane punctured twice. There exist two homotopy classes of loops $\mathfrak{g}_1=[\gamma_1]$ and $\mathfrak{g}_2=[\gamma_2]$ such ...
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Prove the conditions 1 and 1' is equivalent.

Notation:Let $F_2$ be the free group on generators $x,y$. Definition. Let $\Sigma_{g, n}^{m}$ denote a topological surface with genus $g \geqq 0, n \geqq 0$ punctures, and $m \geqq 0$ boundary ...
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1 vote
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equivalent definition of Grothendieck-Teichmüller group

Notation:Let $F_2$ be the free group on generators $x,y$. Denote the profinite completion of group $G$ by $\hat{G} = \underset{N}{\varprojlim} G/N$. For any group homomorphism $$ \begin{aligned} &\...
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1 vote
1 answer
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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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1 vote
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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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2 votes
1 answer
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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 ...
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