one-prime power hypothesis for conjugacy class sizes

p-divisibility of conjugacy class sizes and normal p-complements

LetN be a normal subgroup of a groupG and let p be a prime. We prove that if the p-part of jx j is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement.

Fixed Point Spaces, Primitive Character Degrees and Conjugacy Class Sizes

Nonsolvable Groups Satisfying the One-Prime Hypothesis

Throughout this paper, G is a finite group and Irr(G) is the set of irreducible characters of G. We are particularly interested in the values these characters take on the identity of G. If χ ∈ Irr(G), then χ(1) is the degree of χ. The set of all degrees for G is written cd(G) = {χ(1) |χ ∈ Irr(G)}. In recent years, there has been much interest in finding connections between the structure of a fi...

Normal subgroups and class sizes of elements of prime power order∗

If G is a finite group and N is a normal subgroup of G with two Gconjugacy class sizes of elements of prime power order, then we show that N is nilpotent.

Characterization of $mathrm{PSL}(5,q)$ by its Order and One Conjugacy Class Size

Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{| mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.

Groups with one conjugacy class of non-normal subgroups‎ - ‎a short proof

For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. We give a short proof of a theorem of Brandl, which classifies finite groups with $nu(G)=1$.