Sizings of groups
by
Thomas Tucker
Colgate University

A function s that assigns a nonnegative integer to each finite group we call a sizing if s(G) is less that or equal to s(H) whenever G is a subgroup of H. The order of a group is obviously a sizing. Other examples include the genus of a group, the symmetric genus of a group, the least n such that G acts faithfully on n symbols, the Albertson-Boutin isometry dimension of G, the least n such that G is a subgroup of the Galois group for a polynomial of degree n, the number of subgroups of G. Nonexamples include the minimal number of generators of G, the order of the abelianization of G, the number of normal subgroups of G. Motivated by the genus of a group, we consider various questions one might ask about a sizing s: for example,whether the number of groups of a given size is 0, 1, or finite, or how s compares aymptotically to the order of a group.

Date received: November 3, 2000