Many Frobenius complements have even order

Abstract

Frobenius groups play a significant role in the theory of finite groups, and, in particular, in the study of simple groups. For example it is not unusual in finite group theory that a proof analyzing a minimal counterexample will end up considering Frobenius groups. Every Frobenius group G is a semidirect product K⋊H where K is a canonical normal subgroup of G (the Frobenius kernel of G) and H is a subgroup of G (a Frobenius complement of G) [1, 35.25(1)]. The theory of Frobenius groups with abelian Frobenius kernel largely reduces to algebraic number theory and indeed to a tractable part of algebraic number theory: the study of unramified primes in abelian extensions of the field of rational numbers [2]. For example it is fairly easy to count the exact number of isomorphism classes of such groups of order less than 10; there are 569, 342 of them [2, p. 85]. It is natural then to ask if it is common for Frobenius groups to have abelian kernel. QUESTION. Do almost all (or even a positive proportion of) Frobenius groups have abelian Frobenius kernel? When the complement of a Frobenius group has even order, then it is known that the kernel is abelian [3, Theorem 3.4A]. About 88.8% of Frobenius complements of order at most 10 have even order [2, p. 54].