On Supersolvable Groups and the Nilpotator

Abstract

A finite group G is called G a 𝒯-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ-subgroup of G if N _G (K) ∩ K ^g  ⊆ K for all g ∈ G. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either 𝒯-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋ- subgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.

Key Words:

• Supersolvable groups
• 𝒯-groups
• ℋ-groups
• The nilpotator and the supersolvator of a group

Acknowledgments

The first author was partly supported by the Hungarian National Foundation for Scientific Research, Grants No. T 038059 and T 034878.

The second author is grateful to the Department of Mathematics of the University of Hawaii at Manoa for its hospitality and support, while this investigation was carried out.

Notes

^#Communicated by A. Turull.