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 all elements outside which have order p; (c) to be a semidirect product P = P1 ⋊ < 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 A of automorphisms 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| (Theorem 1). The proof is based on a similar result derived by the author and P. V. 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 shown that if a finite p-group P with a partition admits an automorphism group A that acts faithfully on P/Hp(P), then the exponent of P is bounded in terms of the exponent of CP (A) (Theorem 2). The proof of this result has its basis in the author’s positive solution of an analog of the restricted Burnside problem for finite p-groups with a splitting automorphism of order p. Both theorems yield corollaries for finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.