Problems of bounding the $p$-length and Fitting height of finite soluble groups

This paper is a survey of some open problems and recent results about bounding the Fitting height and $p$-length of finite soluble groups. In many problems of finite group theory, nowadays the classification greatly facilitates reduction to soluble groups. Bounding their Fitting height or $p$-length can be regarded as further reduction to nilpotent groups. This is usually achieved by methods of representation theory, such as Clifford's theorem or theorems of Hall–Higman type. In some problems, it is the case of nilpotent groups where open questions remain, in spite of great successes achieved, in particular, by using Lie ring methods. But there are also important questions that still require reduction to nilpotent groups; the present survey is focused on reduction problems of this type. As examples, we discuss finite groups with fixed-point-free and almost fixed-point-free automorphisms, as well as generalizations of the Restricted Burnside Problem. We also discuss results on coset identities, which have applications in the study of profinite groups. Finally, we mention the open problem of bounding the Fitting height in the study of the analogue of the Restricted Burnside Problem for Moufang loops.