On some metabelian 3-groups and applications I

Abstract

Let G be a 3-class group of maximal class, and γ₂(G) = [G,G] its derived group. Assume that the commutator factor group G ∕ γ₂(G) is of type (3,3) and the transfers V_{χ2(G) → γ2(G)} and V_{H→ γ2(G)} are trivial, where χ₂(G) is the biggest subgroup of G such that [χ₂(G), γ₂(G)] ⊆ γ₄(G), and, H is one of its maximal normals subgroups different to χ₂(G). Then G is completely determined with the isomorphism class groups of maximal class defined by B.Nebelung in [24]. Moreover the group G is realised. At the end numerical examples illustrating the results are given.