Model theory of finite and pseudofinite groups

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory

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 ⋊ , 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

Rank and order of a finite group admitting a Frobenius group of automorphisms

Suppose that a finite group G admits a Frobenius group FH of automorphisms of coprime order with kernel F and complement H. For the case where G is a finite p-group such that G = G, F, it is proved that the order of G is bounded above in terms of the order of H and the order of the fixed-point subgroup C G(H) of the complement, while the rank of G is bounded above in terms of |H| and the rank of C G(H). Earlier, such results were known under the stronger assumption that the kernel F acts on G fixed-point-freely. As a corollary, for the case where G is an arbitrary finite group with a Frobenius group FH of automorphisms of coprime order with kernel F and complement H, estimates are obtained which are of the form|G| ≤ |C G (F)| · f(|H|, |C G (H)|) for the order, and of the form r(G) ≤ r(C G (F)) + g(|H|, r(C G (H))) for the rank, where f and g are some functions of two variables. © 2013 Springer Science+Business Media New York

On the influence of fixed point free nilpotent automorphism groups

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that for all nonidentity elements . Let FH be a Frobenius-like group with complement H of prime order such that is of prime order. Suppose that FH acts on a finite group G by automorphisms where in such a way that In the present paper we prove that the Fitting series of coincides with the intersections of with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of by at most one. As a corollary, we also prove that for any set of primes , the upper -series of coincides with the intersections of with the upper -series of G, and the - length of G exceeds the -length of by at most one