  We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the
Fitting height and insoluble length of finite groups. As a by-product of our
methods, we also prove a generalization of a result of Flavell, which itself
generalizes Wielandt's Zipper Lemma and provides a characterization of
subgroups contained in a unique maximal subgroup. We also derive a number of
consequences of our theorems, including some applications to the set of odd
order elements of a finite group inverted by an involutory automorphism.

Group Theory