\(\bigvee\)-complete epimorphisms between lattices of normal subgroups (Q1109134)

If \({\mathfrak L}\) and \({\mathfrak L}'\) are \(\bigvee\)-complete lattices, a homomorphism \(\phi\) : \({\mathfrak L}\to {\mathfrak L}'\) is said to be \(\bigvee\)- complete if \(\phi\) (\(\bigvee_{i\in I}x_ i)=\bigvee_{i\in I}\phi (x_ i)\) for each family \((x_ i)_{i\in I}\) of elements of \({\mathfrak L}\). \textit{F. de Giovanni} and the reviewer [Rend., Sci. Mat. Appl., A 116(1985), 45-53 (1982; Zbl 0589.20014)] considered \(\bigvee\)-complete epimorphisms between lattices of normal subgroups of groups. In particular they proved that, if there exists a \(\bigvee\)-complete epimorphism from the lattice n(G) of normal subgroups of the nilpotent p- group G of class 2 onto the lattice n(H) of normal subgroups of the soluble group H, then either H is a hypercentral p-group or it has prime order. In this paper the authors obtain some more information on this problem, proving that, if G is a nilpotent p-group of class 2 whose centre is not locally cyclic and \(H\neq 1\) is a group for which there exists a \(\bigvee\)-complete non-injective epimorphism n(G)\(\to n(H)\), then H and \(G'=Z(G)\) are elementary abelian p-groups, and each normal subgroup of G either is contained in G' or contains G'. Moreover, it is proved that, if Z(G) is locally cyclic, then either H is simple or n(H)\(\simeq n(G/N)\), where N is the unique minimal normal subgroup of G.