A classification of groups satisfying the converse of Lagrange's theorem
This paper is a classification of finite groups satisfying the converse of Lagrange's Theorem. We begin by showing a series of inclusions of classes of finite groups: p-groups {dollar}\subseteq{dollar} nilpotent {dollar}\subseteq{dollar} supersolvable {dollar}\subseteq{dollar} polycyclic {dollar}\subseteq{dollar} solvable. The crucial point of the paper consists of the proof that the class of supersolvable groups is contained in the class of converse Lagrange groups while the class of polycyclic groups is not. We also show that finite cyclic groups and finite abelian groups are included in the class of converse Lagrange groups. Finally, we give an example to show that the class of converse Lagrange groups is not contained in the class of supersolvable groups.
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