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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 \(Gal(F_p^2(K) | K)\) 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.