Generalizations of Groups in which Normality Is Transitive

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 Gaschütz , W. ( 1957 ). Gruppen, in denen das normalteilersein transitiv ist . J. Reine Angew. Math. 198 : 87 – 92 .[Crossref] , [Google Scholar]), Zacher (1964 Zacher , G. ( 1964 ). I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali . Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 37 : 150 – 154 . [Google Scholar]), and Agrawal (1975 Agrawal , R. K. ( 1975 ). Finite groups whose subnormal subgroups permute with all Sylow subgroups . Proc. Amer. Math. Soc. 47 : 77 – 83 .[Crossref], [Web of Science ®] , [Google Scholar]). 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.