In [LP] the following definition was given:

Definition. LetK be a field, G a finite group. Let M be a YLG-module. Let N(G,M) be the (normal) subgroup generated by the elements of G that fix a nontrivial vector in M. Then G/N(G,M) is called a generalized Frobeniue complement (in short: GFC) forG (with respect to M).

Remark.Via a well known theorem by Wielandt, A. Espuelas has shown in [E] that in the more general situation of a finite group G acting on a group Q such that all the elements outside a proper normal subgroup N of G act fixedpoint-freely on Q, G/N is actually isomorphic to a factor group of some GFC for G. Two natural problems arise:

Problem 1. Given a class of groups, study the properties of a GFC for a group G in the class.

Problem 2. Given a finite group H, find a group G such that H is isomorphic to a generalised Frobenius complement for G.

The aim of this note is to state some answers to Problems 1 and 2, when G is a finite p-group. This restriction is a natural starting point, since in [LP] it is shown that the Sylow p-subgroups of a GFC for an arbitrary finite group G are factor groups of suitable GFCs for the Sylow psubgroups of G. As in [LP], and without loss of generality, it is assumed throughout that K = c.