Many Frobenius complements have even order

TL;DR

This paper investigates Frobenius groups with even order complements, showing that under certain prime divisibility constraints, the proportion of such groups with odd order is very small, less than half.

Contribution

It provides bounds on the proportion of Frobenius groups with even order complements based on prime divisibility constraints, advancing understanding of their algebraic structure.

Findings

Proportion of Frobenius complements with odd order is less than 1/2^s.

A positive lower bound on the proportion is established.

The analysis reduces to algebraic number theory techniques.

Abstract

The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound $s$ on the number of distinct primes dividing the order of their commutator subgroups, then the proportion of these with odd order is less than $1/2^{s}$. A positive lower bound is also given.