Disposition p-groups

摘要

For any prime p and positive integers c, d there is up to isomorphism a unique p-group of least order having any (finite) p-group G with rank and Frattini class as epimorphic image. Here is the least positive integer such that G has a central series of length n with all factors being elementary. This "disposition" p-group has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for the centralizer whenever is outside the Frattini subgroup, and that for odd p and the group is a distinguished Schur cover of G with . We also have a fibre product construction of in terms of which might be of interest for Galois theory.