Transfers of metabelian p-groups

Abstract

Explicit expressions for the transfers V i from a metabelian p-group G of coclass cc(G) = 1 to its maximal normal subgroups M 1, . . . , M p+1 are derived by means of relations for generators. The expressions for the exceptional case p = 2 differ significantly from the standard case of odd primes p ≥ 3. In both cases the transfer kernels Ker(V i ) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group \({{\rm{Gal}}({F}_p^2(K)\vert K)}\) of the second Hilbert p-class field \({{F}_p^2(K)}\) of an algebraic number field K. For certain metabelian 3-groups G with abelianisation G/G′ of type (3, 3) and of coclass cc(G) = r ≥ 3, it is shown that the principalisation type determines the position of G on the coclass graph \({\mathcal{G}(3,r)}\) in the sense of Eick and Leedham-Green.