ABSTRACT
ABSTRACT: The various genus parameters for finite groups can be viewed in a broader context. A sizing is a function s from the set of all finite groups to the nonnegative integers satisfying s(A) ≤ s(B) whenever A is isomorphic to a subgroup of B. Numerous example and non-examples are given. Natural questions about a sizing s are its range (gaps), whether s(Q) ≤ s(A) when Q is a quotient group of A, whether s provides a certificate for isomorphism so that s(A) = s(B) implies A is isomorphic to B. Given two sizings, the comparison set C(s, t) is defined to be the set of all rational numbers s(A)/t(A) where t(A) = 0. This provides a general setting for a variety of results like the 5/8 theorem for commuting pairs or a similar 3/4 theorem for proportion of involutions. It also provides a setting for asymptotic comparisons: define s and t to be asymptotic if both C(s, t) andC(t, s) are bounded. It is shown, for example, that all the genus parameters are asymptotic with each other for all groups of genus greater than one, but none are asymptotic to the order of a group.
1 INTRODUCTION
