Related papers: Primitivity rank for random elements in free group…

A group $G$ is called residually finite if for every non-trivial element $g \in G$, there exists a finite quotient $Q$ of $G$ such that the element $g$ is non-trivial in the quotient as well. Instead of just investigating whether a group…

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of…

We show that, if $H$ is a random subgroup of a finitely generated free group $F_k$, only inner automorphisms of $F_k$ may leave $H$ invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally…

The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $\Pi(x;g)$ of primes $p\le x$ for which $g$ is a…

For an epimorphism pi of the free group F_n onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_n for which pi*f = pi. This is called the standard congruence subgroup of Aut(F_n) associated to G and pi. In the…

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…

We develop a refinement of Whitehead's algorithm for primitive words in a free group. We generalize to subgroups, establishing a strengthened version of Whitehead's algorithm for free factors. We make use of these refinements in proving new…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…

Let $G$ be a finite group and let $\pi$ be a set of primes. Write $\mathrm{Irr}_{\pi'}(G)$ for the set of irreducible characters of degree not divisible by any prime in $\pi$. We show that if $\pi$ contains at most two prime numbers and the…

Let $n\geq 3$ be an integer. Let $F_n$ be the Fermat curve defined by the Fermat equation $x^n+y^n=z^n$. For a curve $C/\mathbb{Q}$, we say an algebraic point $P\in C(\bar{\mathbb{Q}})$ is primitive if the Galois group of the Galois closure…

Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{1}{2} are finite. We prove the existence of a uniform quantifier elimination procedure for formulas of minimal rank (probably the superstable part of the…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. An element $\alpha \in \mathbb{F}_{q^n}$ is called $k$-normal over $\mathbb{F}_q$ if $\alpha$ and its conjugates generate a vector subspace of $\mathbb{F}_{q^n}$ of dimension…

We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of…

Let $G$ be a finite group, let $\pi(G)$ be the set of prime divisors of $|G|$ and let $\Gamma(G)$ be the prime graph of $G$. This graph has vertex set $\pi(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$.…

Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $\alpha$ of the finite field $\mathbb{F}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative…

Let D denote the (n-1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of D obtained by starting with the full 1-skeleton of D and then adding each 2-simplex independently with probability p. For a fixed c>0 it is shown that…

We prove that the group algebra $KG$ of a group $G$ over a field $K$ is primitive, provided that $G$ has a free subgroup with the same cardinality as $G$, and that $G$ satisfies the following condition $(\ast)$: for each subset $M$ of $G$…

Let \phi be an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under \phi. The nonzero eigenvalues of \phi are contained in the eigenvalues of \phi restricted to H.

Let $F$ be a finitely generated free group, and let $H\le F$ be a finitely generated subgroup. An equation for an element $g\in F$ with coefficients in $H$ is an element $w(x)\in H*\langle x \rangle$ such that $w(g)=1$ in $F$; the degree of…