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 $M_i$ $(1\le i\le p+1)$ 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 $Gal(F_p^2(K)\mathrm{\mid }K)$ 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 $G\mathrm{/}{G}^{\mathrm{\prime }}$ 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.