Generalizations of Groups in which Normality Is Transitive

Abstract

A group G is called a Hall_𝒳-group if G possesses a nilpotent normal subgroup N such that G/N′ is an 𝒳-group. A group G is called an 𝒳_o-group if G/Φ(G) is an 𝒳-group. The aim of this article is to study finite solvable Hall_𝒳-groups and 𝒳_o-groups for the classes of groups 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯. Here 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯 denote, respectively, the classes of groups in which normality, permutability, and Sylow-permutability are transitive relations. Finite solvable 𝒯-groups, 𝒫𝒯-groups, and 𝒫𝒮𝒯-groups were globally characterized, respectively, in Gaschütz (1957), Zacher (1964), and Agrawal (1975). Here we arrive at similar characterizations for finite solvable Hall_𝒳-groups and 𝒳_o-groups where 𝒳 ∈ {𝒯, 𝒫𝒯, 𝒫𝒮𝒯}. A key result aiding in the characterization of these groups is their possession of a nilpotent residual which is a nilpotent Hall subgroup of odd order. The main result arrived at is Hall_𝒫𝒮𝒯 = 𝒯_o for finite solvable groups.

Key Words:

• Permutable
• S-permutable
• Solvable groups
• Supersolvable groups
• 𝒯-groups

Mathematics Subject Classification:

• 20Dxx
• 20D10
• 20D20
• 20D35

ACKNOWLEDGMENTS

The contents of this article mainly come from the first part of the author's Ph.D. thesis. The author is greatly indebted to the University of Kentucky and in particular to the author's Ph.D. advisor, Dr. Jim Beidleman. Without Dr. Beidleman's guidance, this work would have not been possible.

The author would also like to express his thanks to the referee for the careful reading of the article and the suggestions that have lead to a better presentation.

Notes

¹An Iwasawa group is one in which every subgroup is permutable.

Communicated by M. R. Dixon.