Transfers of metabelian p-groups

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 a parts per thousand yen 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 of the second Hilbert p-class field 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 a parts per thousand yen 3, it is shown that the principalisation type determines the position of G on the coclass graph in the sense of Eick and Leedham-Green.