FINITE GROUPS ALL OF WHOSE MAXIMAL SUBGROUPS ARE SB-GROUPS

Abstract

Abstract. A finite group G is called a SB-group if every subgroup ofG is either s-quasinormal or abnormal in G. In this paper, we give acomplete classification of those groups which are not SB-groups but allof whose proper subgroups are SB-groups. 1. IntroductionAll groups considered here are assumed to be finite, and all notations em-ployed are standard.Let Σ be an abstract group theoretical property. If all proper subgroups ofa group G have the property Σ but G does not have the property Σ, then Gis called a minimal non-Σ-group. One of the hottest topics in group theory isto determinate the structure of minimal non-Σ-groups and many meaningfulresults about this topic have been obtained. For example, O. J. Schmidt [12]determined the structure of minimal non-nilpotent groups, K. Doerk [4] eluci-dated the structure of minimal non-supersolvable groups, and J. G. Thompson[14] gavethe classificationofnonsolvable groupsall of whoselocal subgroupsaresolvable. These achievements have greatly pushed forward the developments ofgroup theory. The more new results can be referred to refs. [7, 8, 9, 11, 13, 17].The aim of the present article is to study the structures of some minimal non-Σ-groups. Recall that a subgroup H of a group G is an abnormal subgroupif x ∈ hH,H