In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on Fq of the form c 7→ act , where q is a power of 2, t is an integer with 0 < t ≤ q1/4, and a ∈ Fq. This sett...

Let K be a non-archimedean field, and let φ ∈ K(z) be a rational function of degree d ≥ 2. If φ has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that φ is conjugate over L to a map of good reduction. In particular, if d = 2 or d is less than the residue characteristic of K, the bound is d + 1. If K is discretely valued,...

For a prime p, we consider some natural classes of matrices over a finite field Fp of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [−H,H] ⊆ [−(p ...