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.

22 pages, 7 tables

References in corpus (1)

• The second p-class group of a number field

Cited by in corpus (7)

• The second p-class group of a number field
• Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture
• Index-\(p\) abelianization data of \(p\)-class tower groups
• The distribution of second p-class groups on coclass graphs
• The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields
• Normal lattice of certain metabelian p-groups G with \(G/G^\prime\simeq (p,p)\)
• Periodic bifurcations in descendant trees of finite (p)-groups