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 $F_n$.
199
questions with no upvoted or accepted answers
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Computing $\mathrm{Fix}(\phi)$ for autormophisms $\phi$ of free groups
Let $F_A$ be the free group generated by the finite set $A$ and let $\phi\colon F_A \to F_A$ be a group-automorphism. It is known [1] that
$$ \mathrm{Fix}(\phi) = \{g \in F_A : \phi(g) = g\} $$
is (...
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votes
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Identifying a certain subgroup of a free group
Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ ...
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Is there a sense in which $(\mathbb{R},+)$ is "free"?
Given a set $S$, the free group on $S$ consists of finite strings of elements in $S$. They can be visualized as paths on the integer grid $\mathbb{Z}^S$ starting at the origin, with the group ...
user519413
5
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135
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Question about the computation of a formal power series
Consider the group ring $A$ of a free finitely generated group (i.e. noncommutative Laurent polynomials) with coefficients in a field $\mathbf{k}$ of characteristic $0$.
Denote by $\tau: \: A \...
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5
votes
3
answers
821
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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 $F_a$ ...
- 71
5
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Group generated by two polynomials
The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of $Homeo(\mathbb{R})...
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$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 $M_1=\left(\begin{array}{ccc}1 &0&0\\0&\frac 35&-\frac{4}5\\...
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A Conjecture in Low-Dimensional Topology.
Context
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
4
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Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(\mathbb F_d,\...
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Recovering an element of a free group from its projections
Assume you have an unknown word on an alphabet with at least three letters, and you know all the words obtained by erasing each copy of some letter. Then, you can find the first letter of the original ...
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4
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An algorithm determining whether two subgroups of a free group are automorphic
In the book Lyndon, Schupp, Combinatorial Group Theory, the edition from 2000 P.30 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two subgroups ...
- 701
4
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Two rotations in $\mathbb R^3$ by irrational angle generate free group $F_2$.
Let $A,B\in $ SO$(3)$ so that $A, B$ are rotations by an angle $\theta\in \mathbb R\setminus \mathbb Q$ about the $z$-axis and $x$-axis, respectively. I want to prove that the group $\langle A, B\...
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Do quasi-isometries of semi direct products of free groups preserve fibres
Is it true that any quasi-isometry between semi direct products $Z^{n} \times_{f} F, Z^{n} \times_{f'} F$ for $F$ finitely generated free non-abelian group (for $F$ abelian it is obviously false) ...
- 141
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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 $[G:N] < \infty$. Prove that
$$
\bigcap_{N \in \mathcal{N}} N \ = \ \{ e\}
$$
I know how free ...
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4
votes
1
answer
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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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