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_i) ((1\le i\le 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\ge 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 (Gal(F_p^2(K) | 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^{\prime}) of type (3,3) and of coclass (cc(G)=r\ge 3), it is shown that the principalisation type determines the position of G on the coclass graph G(3,r) in the sense of Eick and Leedham-Green.