Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group ({G_{d}^{c}(p)}) of least order having any (finite) p-group G with rank ({d(G) le d}) and Frattini class ({c_{p}(G) le c}) as epimorphic image. Here ({c_{p}(G) = n}) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group ({G_{d}^{c}(p)}) 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 ({G = G_{d}^{c}(p)}) the centralizer ({C_{G}(x) = langle Z(G), x rangle}) whenever ({x in G}) is outside the Frattini subgroup, and that for odd p and ({d ge 2}) the group ({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}) is a distinguished Schur cover of G with ({E/Z(E) cong G}). We also have a fibre product construction of ({G_{d}^{c+1}(p)}) in terms of ({G = G_{d}^{c}(p)}) which might be of interest for Galois theory.