In 1903, Ferdinand Georg Frobenius made a conjecture that can be stated as such: let G be a group that emits a fixed-point-free automorphism. Then, G is a solvable group. Throughout the rest of the 20th century many different specific cases of this conjecture have been proved (with the cases putting a restriction on the order of the automorphism). For example, in 1959 John Thompson proved this for all such automorphisms with prime order. Later on in the 70's, Elizabeth Ralston proved this result for autmorphisms with the order being a product of two distinct primes for two primes.
Finally in the 90's the full conjecture was accepted as being proven as a consequence of some results in the landmark Classification of Finite Simple Groups. As a result, an attempt at an all-encompassing and unified proof of this conjecture has been largely abandoned by group theorists. For this thesis, we will look at a specific case of this conjecture where the order of the automorphism is 4 (as also done in [5]) and try to give a formal proof whilst introducing the necessary results used as tools in said proof.