A dual version of Huppert's conjecture on 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

one-prime power hypothesis for conjugacy class sizes

a finite group $g$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. taeri conjectured that an insoluble group satisfying this condition is isomorphic to $s times a$ where $a$ is abelian and $s cong psl_2(q)$ for $q in {4,8}$. we confirm this conjecture.

A NOTE ON THE COMMUTING GRAPHS OF A CONJUGACY CLASS IN SYMMETRIC GROUPS

The commuting graph of a group is a graph with vertexes set of a subset of a group and two element are adjacent if they commute. The aim of this paper is to obtain the automorphism group of the commuting graph of a conjugacy class in the symmetric groups. The clique number, coloring number, independent number, and diameter of these graphs are also computed.

A Conjecture about Conjugacy in Free Groups

Say that an element of a free group is a pure conjugate if it can be expressed from the generators using exclusively the conjugacy operation. We study free reductions in words representing pure conjugates. Using finite state automata, we attribute to the letters in such words levels that live in some free left distributive system. If a certain conjecture about this system is true, then reductio...