Let σ={σi∣i∈I} be some partition of the set of all
primes P and let G be a finite group. Then G is said to be σ-full if G has a Hall σi-subgroup for all i. A subgroup A of
G is said to be σ-permutable in G provided G is σ-full and
A permutes with all Hall σi-subgroups H of G (that is,
AH=HA) for all i. We obtain a characterization of finite groups G in
which σ-permutability is a transitive relation in G, that is, if K
is a σ-permutable subgroup of H and H is a σ-permutable
subgroup of G, then K is a σ-permutable subgroup of G.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.0250