Automorphisms of finite p-groups admitting a partition

For a finite p-group P the following three conditions are equivalent: (a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial pairwise intersections; (b) to have a proper subgroup outside of which all elements have order p; (c) to be a semidirect product P = P1o ⟨φ⟩ where P1 is a subgroup of index p and φ is a splitting automorphism of order p of P1. It is proved that if a finite p-group P with a partition admits a soluble group of automorphisms A of coprime order such that the fixed-point subgroup CP (A) is soluble of derived length d, then P has a maximal subgroup that is nilpotent of class bounded in terms of p, d, and |A|. The proof is based on a similar result of the author and Shumyatsky for the case where P has exponent p and on the method of “elimination of automorphisms by nilpotency”, which was earlier developed by the author, in particular, for studying finite p-groups with a partition. It is also proved that if a finite p-group P with a partition admits a group of automorphisms A that acts faithfully on P/Hp(P ), then the exponent of P is bounded in terms of the exponent of CP (A). The proof of this result is based on the author’s positive solution of the analogue of Restricted Burnside Problem for finite p-groups with a splitting automorphism of order p. Both theorems yield corollaries on finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.