The distribution of second p-class groups on coclass graphs

Abstract

General concepts and strategies are developed for identifying the isomorphism type of the second p-class group \(G=Gal(F_p^2(K) | K)\), that is the Galois group of the second Hilbert p-class field \(F_p^2(K)\), of a number field K, for a prime p. The isomorphism type determines the position of G on one of the coclass graphs G(p,r), \(r\ge 0\), in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p-class group \(Cl_p(K)\), the position of G is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second p-class groups G for various series of number fields K having p-class groups \(Cl_p(K)\) of fixed type and p in {2,3,5,7}.