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.

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Translated from *Algebra i Logika*, Vol. 51, No. 3, pp. 392-411, May-June, 2012.

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Khukhro, E.I. Automorphisms of finite *p*-groups admitting a partition.
*Algebra Logic* **51, **264–277 (2012). https://doi.org/10.1007/s10469-012-9189-2

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### Keywords

- splitting automorphism
- finite
*p*-group - exponent
- derived length
- nilpotency class
- Frobenius group of automorphisms