Finite groups with invariant TI-subgroups or σ-subnormal subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is called a TI-subgroup of [Formula: see text] if [Formula: see text] or [Formula: see text] for each [Formula: see text] [Formula: see text][Formula: see text]. Furthermore, let [Formula: see text] be a partition of the set of all primes [Formula: see text], a subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-subnormal in [Formula: see text] if there is a chain of subgroups [Formula: see text] where, f