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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Prove that $ \text{Out}(F_2) \cong \text{GL}_2(\mathbb{Z})$
Let $F_2$ be a free group of rank $2$. Define: $\text{Out}(F_2)=\text{Aut}(F_2) / \text{Inn}(F_2)$.
Prove that:
$$ \text{Out}(F_2) \cong \text{GL}_2(\mathbb{Z})$$
In previous exercise, I showed that:
...
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A map from free group into $\mathbb{Z}$ [closed]
Why can a map $F\rightarrow \mathbb{Z}_2 $ from a free group $F$ into $\mathbb{Z}_2$ be lifted to $\mathbb{Z}$?
Suppose $X\subset F$ is such that $F=\langle X\rangle$. Let $x\in X$. Choose $n_x\in \...
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$3$ sheeted covers of $\mathbb{R}P^2 \vee S^1$
Using covering spaces, show that there are three subgroups of index $3$ of $G = \mathbb{Z}_2 * \mathbb{Z}$. This is a problem from a qualifying exam.
Let $X = \mathbb{R}P^2 \vee S^1$. Then, $\pi_1(\...
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Meaning of cut vertex in Whitehead graph in the sense of language
I recently reading Whitehead algorithm in paper of Stallings (without bothering 3-manifolds). Whitehead graph and the cut vertex play an important role. Let us consider one example from Stallings ...
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Writing an element of $\operatorname{Out}(F_n)$ as a composition of generators
Let $F_n = \langle x_1, ..., x_n | \rangle$ be the free group. Consider the following maps in $\operatorname{Out}(F_n)$:
$\sigma_{ij}$ sends $x_i \rightarrow x_i x_j$
$\epsilon_i$ sends $x_i \...
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Showing that $\langle A \rangle$ can be equivalently defined via $F(A)$ and via $F^{ab}(A)$ when $G$ is abelian, $A \subset G$.
In Aluffi's Algebra: Chapter 0 the author introduces the subgroup generated by a subset as follows (paraphrased):
Let $G$ be a group, and $A \subset G$ any subset. By the universal property of free ...
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Is there a general way to find the inverse of an automorphism of the free group? [closed]
If we describe an automorphism of the free group (on n generators) by where it sends the generators, is there some kind of algorithm to find the inverse automorphism? I am particularly interested in ...
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Relative divisibility of derived subgroup of free group [closed]
Let $F$ be a free group (possibly on an infinite set) and let $[F,F]$ denote its derived subgroup.
Can there be a $w \in F \setminus [F,F]$ and an $n>0$ such that $w^n \in [F,F]$?
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Questions on the $\hom$ Functor and Free Groups
This question arises while learning about the $\hom$ functor. My algebra background is not that strong, so here is my question:
Let $G$ be a free group, and let $f\colon G \to G$ be a group
...
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Transition matrix associated to representative of element in $Out(F_n)$
There is a notion of transition matrix associated to elements in $Out(F_n)$ from Bestvina and Handel's paper that I am a little bit confused.
Let $\Phi\in Out(F_n)$ and $\phi:\Gamma\to\Gamma$ a ...
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Every graph morphism that is an immersion and surjective on the fundamental group is a homeomorphism
Here, we consider graphs as 1-dimensional CW complex and a graph morphism is a map sending vertices to vertices and $[f(a),f(b)]=f([a,b])$ where $[a,b]$ represents an edge connecting vertices $a,b$. A ...
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Direct sum of free abelian group and quotient of abelian group by subgroup
I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem:
Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
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$SO_3(\mathbb Q)$ contains a free group using 5-adic numbers
I am trying to show that $SO_3(\mathbb Q)$ contains a free group using 5-adic numbers, and more precisely using the matrices $M_1=\left(\begin{array}{ccc}1 &0&0\\0&\frac 35&-\frac{4}5\\...
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Can we construct a free structure on a non associative algebraic structure.
For any set we can construct a free group on it. Also for non associative structures like Lie algebra, Lie ring we may construct free structures, but these are non associative structures and having ...
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Technique for showing a group is not free?
The specific case I present here is much less important than the general question. I have two matrices: $$
p = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & -\frac{1}{2} \\
0 &...
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"Almost Retractible" Abelianizations of Groups
I have two related questions.
Is there a name for a nonabelian group $G$ whose abelianization is $\bigoplus_{i=1}^n \mathbb{Z}/p_i\mathbb{Z}$ such that for each $i$ there is an element $g$ whose ...
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Tiling of a tree to show that a group acting freely on a tree is free
Let me start giving some context:
Let $G$ be a group acting freely on a tree $T$. Let $T'$ be the barycentric subdivision of $T$ (that is, the graph obtained by placing a new vertex at the center of ...
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If a graph map is an immersion, then the induced homomorphism on fundamental groups is injective
So I was reading some Geometric group theory and came across Stalling's folding of graphs. Now I am trying to use the folding idea to prove that every finitely generated subgroup of a free group is ...
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How to prove the free group $F_n$ cannot be generated by $n$ elements as a monoid?
I cannot figure out how to prove that the submonoid of $F_n$ generated by elements $\alpha_1,\dots , \alpha_n$ will never be the whole group.
It clearly is possible to generate the free group on $a_1, ...
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Free product contains the free product of itself with a free group.
I saw this answer and I'm thinking if with this idea we can show that $G_1 \ast G_2 \ast F$ (where $F$ is a free group of countable rank) embedds in $G_1 \ast G_2$ if $G_1$ or $G_2$ has cardinality at ...
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Sorting integers by looking at their prime factorizations
By the fundamental theorem of arithmetic, we know that any positive integer can be uniquely defined by its prime factors. Now, suppose $S_{\infty}$ is the set of all primes, and let $s_i
\in S$ such ...
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A question about commutators in free groups
Let $F$ be the free group on $X=\{ x_1,\dots, x_n\}$ for some $n\geq2$. Define the lower central series of $F$ inductively: $\gamma_1(F):= F$, $\gamma_{i+1}(F)=[\gamma_i(F),F]$ for $i\geq1$. Is it ...
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Condition on finitely generated subgroup of $GL_2(\mathbb{Q})$ to be free
I am considering finitely generated subgroup $G=\langle A,B,C\rangle$ of $GL_2(\mathbb{Q})$ such that $A,B,C$ all have the upper triangular form
$$A=\begin{pmatrix}a_1 & a_2 \\ 0 & a_3\end{...
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Virtual solvability of dense subgroups
Let $G$ be a (finitely generated) dense subgroup of $\mathsf{SL}(2;\mathbb{C})$. Is it possible that $G$ is virtually solvable?
In other words, by Tit's alternative, does being dense necessitate the ...
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Exercise on Generators and Relations from Michael Artin's book
The question is:
Let $\phi: G \mapsto G'$ be a surjective group homomorphism. Let $S$ be a subset of $G$ whose
image under $\phi$(S) generates $G$', and let $T$ be a set of generators of $\ker\phi$. ...
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Prove a surjective endomorphism $\phi$ of a 1-relator group $ ⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩ $ is not injective
Consider the infinite group $H$ with presentation
$$
⟨a, b ∣ a^{-1} b^2 a b^{-3}⟩
$$
so that the relation is $a^{-1} b^2 a=b^3$.
The map
$$
a ↦ a\\b ↦ b^2
$$
induces a surjective homomorphism $ϕ:H\to ...
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How to show that the trivial group is the free group of the empty set (using universal property of free groups)?
Aluffi (in Algebra: Chapter 0) says that given a set $A$, the free group is a group $F(A)$ together with a set map $j_*:A\to F(A)$ st for any group $G$ and any set map $f:A\to G$, there is a unique ...
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$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point
Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
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Free group on $X$ means no relation in $X^{\pm}$ [closed]
I am reading Free Groups from the book ``Presentations of Groups" by D. L. Johnson.
The author says that the existence of $\theta'$ means there is no relation in $X^{\pm}$. He gives the argument ...
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Finding free subgroup $F_2$ in the free product $\frac{\mathbb{Z}}{5\mathbb{Z}} * \frac{\mathbb{Z}}{6\mathbb{Z}}$
Is there any free group isomorphic to $F_2$ contained in the free product group $\frac{\mathbb{Z}}{5 \mathbb{Z}}* \frac{\mathbb{Z}}{6 \mathbb{Z}}?$
Let $\frac{\mathbb{Z}}{5\mathbb{Z}}= \langle a \mid ...
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Burnside groups with GAP system [closed]
My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP.
The obvious representation using relations (see example for ...
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Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?
I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student ...
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If there is a bijection from a subset $S$ of a group $G$ onto $X$ then $F(X)$ isomorphic to $\langle S \rangle$, Where $F(X)$ free group on X
Let $\phi: G \to F(X)$ be a group homomorphism suppose that $\phi$ maps a subset $S$ of $G$ bijectively onto $X$. Then $F(X) $ is isomorphic to $\langle S\rangle$, where $F(X)$ free group with basis $...
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Free object is a free group in the category of groups
I have a question and would appreciate a clear answer.
Firstly, I will provide an introduction regarding my understanding, and then I will ask my question.
Let's begin with the definition of a ...
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Lee Mosher book definition of a tree.
I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.
I am wondering how is this true, can someone explain this to me please?...
user965463
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Proof of the universal property of free abelian groups
Let $S$ be a set. The group with presentation $(S, R)$ , where $R = \{\ [s , t] \mid s,t\in S\ \}$ is called the free abelian group on $S$ -- denote it by $A (S)$ . Prove that $A (S)$ has the ...
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Banach-Tarski Paradox: Extension with cycles
I am new to StackExchange so apologies if my question is poorly asked or does not abide by the standards.
Referring to the 2016 edition of Tomkowicz and Wagons' book on the Banach-Tarski Paradox, ...
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Does every finitely generated dense subgroup of $ SU(n) $ contain a free subgroup? [closed]
I read in On the spectral gap for finitely-generated subgroups of SU(2) that every finitely generated dense subgroup of $ SU(2) $ contains a free subgroup.
Is it true in general that every finitely ...
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When does the closure of a free subgroup of $\mathsf{SL}(2;\mathbb{C})$ equal $\mathsf{SL}(2;\mathbb{C})$
Let $A \subset \mathsf{SL}(2; \mathbb{C})$ be a finite set of matrices. Consider the set $$S = \overline{\langle A \rangle},$$ where we take the closure in $\mathsf{SL}(2; \mathbb{C})$, and where $\...
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Are right-adjoints of a forgetful functor reflectors?
From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them ...
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Bipartite intersection graph
Recently, I'm trying to learn geodesic currents on free groups through a paper by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form ...
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Blocks for Extending a Primitive System in a Free Group
Let $F$ be a free group of rank at least two, $A$ free a factor of $F,$ and $x$ a primitive element in $F \setminus A.$ Suppose that for some/every basis $\mathcal A$ of $A,$ the set $\mathcal A \cup \...
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Is it possible to produce with $3$ elements the group $G=F_2 \ast (\mathbb{Z} \times \mathbb Z)$?
Question: Is it possible to produce with $3$ elements the group $G=F_2 \ast (\mathbb{Z} \times \mathbb Z)$?
I figured that $G= \langle a,b,c,d \mid [c,d]=1 \rangle$.
I also know that it is impossible ...
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A Conjecture in Low-Dimensional Topology.
Context
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
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Covering space of compact surface with free fundamental group
Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?
I know that if we ...
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Is there an enumeration of finitely presented groups?
I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a ...
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$F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$.
In a paper I read that $F_2\ltimes F_2^{2n-4}$ is a subgroup of $\mathrm{Aut}(F_n)$. The proof of this fact is as follows:
Choose $F_2\leq \mathrm{Aut}(F_2)$ and let it act diagonally on $F_2^{2n-4}$, ...
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Ways to show that words with exponent sum zero for each generator are elements of the commutator subgroup
Say I have a free group on the generators $X = \{ x_1, x_2, ... , x_k \}$ with $k \geq 2$. I read (in an article) that if I have a word $w$ written in the generators and their inverses, and the ...
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Can you determine the order of a generator in this group presentation? [closed]
Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x.
My follow up question: Is there a way to determine the order without finding ...
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Schreier basis of kernel of $F(G)\to G$ for $G$ a group
Let $G$ be a group and $F(G)$ the free group on $G$ as a set. There is a natural epimorphism $F(G)\to G$ that maps $[\sigma]\in F$ to $\sigma$, let $K$ be its kernel. Is the set $$X=\{[\sigma][\tau][\...
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