On the co-Dedekindian finite p-groups with non-cyclic abelian second centre

Abstract

Abstract. A group Gis called co-Dedekindianif every subgroup ofGis invar-iant under all central automorphisms of G. In this paper we give some necessaryconditionsforcertainfinitep-groupswithnon-cyclicabeliansecondcentretobeco-Dedekindian.Wealsoclassify3-generatorco-Dedekindianfinitep-groupswhichareofclass3,havingnon-cyclicabeliansecondcentrewithj 1 ðG p Þj ¼ p.2000MathematicsSubjectClassification.20E34,20D15,20D45.1. Introduction. Let Gbe a group, and let ZðGÞ denote the centre ofG.Anautomorphism of Gis called centralif x 1 ðxÞ2ZðGÞ for each x2 G. The set ofall central automorphisms of G, denoted by Aut c ðGÞ, is a normal subgroup of thefull automorphism group of G. A group Gis called co-Dedekindian(C-group forshort) if every subgroup of Gis invariant under all central automorphisms of G.In [1], Deaconescu and Silberberg give a Dedekind-like structure theorem for thenon-nilpotent C-groups with trivial Frattini subgroup and by reducing the finitenilpotentC-groupstothecaseofp-groupstheyobtainthefollowingtheorem.Theorem 1.1.LetGbeap-group.IfGisanon-abelianC-group,thenZ