Abstract
Let G be a finite group and be a partition of the set of all primes
that is,
and
for all
A set
of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of
is a Hall σi-subgroup of G for some i and
contains exactly one Hall σi-subgroup of G for every
A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain
such that either
is normal in Ai or
is σ-primary for all
σ-abnormal in G if
is not σ-primary whenever
and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set
such that for any
G has a subgroup A of order
and A is σ-abnormal or σ-subnormal in G.