Carlo M. Scoppola

GENERALIZED FROBENIUS GROUPS. II

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |CG(x)|=|CG/K(xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.

Applications of dimension subgroups to the power structure ofp-groups

Dimension subgroups in characteristicp are employed in the study of the power structure of finitep-groups. We show, e.g., that ifG is ap-group of classc (p odd) andk=⌜logp((c+1)/(p−1))⌝, then, for alli, any product ofpi+k th powers inG is api th power. This sharpens a previous result of A. Mann. Examples are constructed in order to show that our constantk is quite often the best possible, and in any case cannot be reduced by more than 1.

Abelian generalized Frobenius complements forp-groups and the Hughes problem

1. Let G be a finite group acting on a group M, and let N(G,M) be the subgroup of G generated by all the elements of G that fix a nontrivial element of M (wherever G and M are specified by the context without possibilities of confusion we wilt write simply N for N(G, M)). In [6], the factor group G/N above is called a generalized Frobenius complement (GFC, in short) for G; in [1], each factor group of G/N is called a Frobenius-Wielandt complement. In this paper the attention is focused on GFCs for finite p-groups acting over complex vector spaces. The restriction to p-groups is suggested by the remark that the first problem, when studying the structure of G/N, is the determination of the properties of its Sylow subgroups: in some cases this information is enough to describe easily GIN. For example a description of the structure of GIN when the Sylow subgroups of G are regular (in the sense of P. Hall) is obtained in [6] and in [7] as a consequence of the following:

Soluble normally constrained pro-p-groups

Abstract A pro-p-group G is said to be normally constrained (or, equivalently, of obliquity zero) if every open normal subgroup of G is trapped between two consecutive terms of the lower central series of G. In this paper infinite soluble normally constrained pro-p-groups, for an odd prime p, are shown to be 2-generated. A classification of such groups, up to the isomorphism type of their associated Lie algebra, is provided in the finite coclass case, for p > 3. Moreover, we give an example of an infinite soluble normally constrained pro-p-group whose lattice of open normal subgroups is isomorphic to that of the Nottingham group. Some general results on the structure of soluble just infinite pro-p-groups are proved on the way.