Self-contained review of Thompson's fixed-point-free automorphism theorem

Abstract

In the early 1900s, Frobenius conjectured if a group [italic G] admits a fixed-point-free automorphism [italic Phi], then [italic G] must be solvable. During the next half-century, mathematicians would struggle to find a completely group theoretic proof of Frobenius' Conjecture. Between 1960 and 1980, progress was made on the Conjecture only by assuming conditions on the order of [italic Phi]. In 1959, Thompson proved, for his dissertation, the case assuming the automorphism had prime order and resulted in a stronger condition than solvable [Tho59]; Hernstein and Gorenstein proved the conjecture with an automorphism of order 4 [DG61]; and in 1972, Ralston proved a group admitting a fixed-point-free automorphism with order [italic pq] is solvable, where [italic p] and [italic q] are primes. [Ral72] It was not until the 1980s, with the power of the Classification of Finite Simple Groups, was Frobenius' Conjecture finally proven; however, the proof involved character theory. In this paper, we consider John Thompson's case of the Frobenius Conjecture: Theorem ([Tho59]). Let G be a group admitting a fixed-point-free automorphism of prime order. Then G is nilpotent. Our goal is to lay a complete framework of the necessary concepts and theorems leading up to, and including, the proof of Thompson's theorem.