Related papers: Powers of sets in free groups

We prove in this note that, for an alphabet with three letters, the set of first return to a given word in a set satisfying the tree condition is a basis of the free group.

We give a simple proof of the well-known fact: any group of n elements is cyclic if and only if n and \phi(n) are coprime. This note is accessible for students familiar with permutations and basic number theory. No knowledge of abstract…

We show that for a finite, nonempty subset $A$ of a group, the quotient set $A^{-1}A:=\{a_1^{-1}a_2\colon a_1,a_2\in A\}$ has size $|A^{-1}A|\ge\frac53\,|A|$, unless $A$ is densely contained in a coset, or in a union of two cosets of a…

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true…

Let $\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. Suppose that for any $G\in\mathcal C$ whose order is divisible by at most two distinct primes there exists an…

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of…

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of…

The Hanna Neumann conjecture states that if F is a free group, then for all finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [ rank(H)-1 ] [ rank(K)-1 ] In this paper, we show that if one of the subgroups, say H, has a…

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th…

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which…

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial…

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n,…

In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over $\Q (\sqrt{-d})$, $d \equiv 7 \pmod 8$ positive and square free, such that $< u^ n,v^n>$ is free for some $n\in \mathbb{N}$. Here we extend this…

We classify groups generated by powers of 2 Dehn twists which are 1) free or 2) have no ``unexpected'' reducible elements. We give some sufficient conditions in the case of groups generated by powers of more than two twists.

We write $S_{\leq n}(A)$ and $\Part_{\fin}(A)$ for the set of permutations with at most $n$ non-fixed points, where $n$ is a natural number, and the set of partitions whose members are finite, respectively, of a set $A$. Among our results,…

We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F…

We present a proof of the following claim. Suppose that $n$ is an integer such that $n>1$ and that $k$ is any field. Suppose that $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order. Then the set $\{h\in\mathrm{SL}(n,k)\mid <g,h>$ is…

Factors $\frac{X}{Y}$ in a free group $F$ with $Y$ normal in $X$ are considered. Precise results on the free structure of ${Y}$ relative to the free structure of ${X}$ when $\frac{X}{Y}$ is abelian are obtained. Some extensions and…

For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.

Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of finite rank which is not equal to 1, or to 3, then the automorphism group Aut(N) of N is complete. The main result of the present paper states that the…