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Transfers of metabelian p-groups

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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.

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  1. Naglergasse 53, 8010, Graz, Austria

    Daniel C. Mayer

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  1. Daniel C. Mayer
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Correspondence to Daniel C. Mayer.

Additional information

Communicated by John S. Wilson.

Research supported by the Austrian Science Fund, Grant Nr. J0497-PHY.

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Mayer, D.C. Transfers of metabelian p-groups. Monatsh Math 166, 467–495 (2012). https://doi.org/10.1007/s00605-010-0277-x

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