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 = P ₁ ⋊ < φ>, where P ₁ is a subgroup of index p and φ is a splitting automorphism of order p of P ₁. 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 C _P (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/H _p (P), then the exponent of P is bounded in terms of the exponent of C _P (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.