Many Frobenius complements have even order

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.

Keywords:

• Distribution of primes
• Frobenius complement
• Frobenius group
• Frobenius triple
• Sylow-cyclic group

2020 Mathematics Subject Classification:

• Primary: 20D60
• Secondary: 11N13
• 20E99