An Arithmetical Condition on the Sizes of Conjugacy Classes of a Finite Group

An element [Formula: see text] of a finite group [Formula: see text] is said to be primary if the order of [Formula: see text] is a prime power. We define [Formula: see text] as follows: if [Formula: see text] is a prime power for every primary element [Formula: see text] of [Formula: see text], where [Formula: see text] is the conjugacy class of [Formula: see text] in [Formula: see text], then [Formula: see text]; if there exists a primary element [Formula: see text] in [Formula: see text] such that [Formula: see text] is divisible by at least two distinct primes, then [Formula: see text]. In this paper we discuss the influence of the number [Formula: see text] on the structure of [Formula: see text].