Derived length and 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.

Groups Whose Nonlinear Irreducible Characters Separate Element Orders or Conjugacy Class Sizes

A class function φ on a finite group G is said to be an order separator if, for every x and y in G\{1} , φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G \ {1} , φ(x) = φ(y) is equivalent to |CG(x)| = |CG(y)| . In this paper, finite groups whose nonlinear irreducible complex characters are all order separa...

Scorecard construction with unbalanced class sizes

A long-running issue in scorecard construction in retail banking is how to handle dramatically unbalanced class sizes. This is important because, in many applications, the class sizes are very different. We describe the impact ignoring such imbalance can have and review the various strategies which have been proposed for tackling it, embedding them in a common theoretical framework. We then des...