Prime Factors of π-Partial Character Degrees and Conjugacy Class Sizes of π-Elements

摘要

Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.