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, for every [Formula: see text], [Formula: see text] is normal in [Formula: see text] or [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. In this paper, we first give a equivalent condition for a group [Formula: see text] in which every self-centralizing subgroup is a TI-subgroup or a [Formula: see text]-subnormal subgroup. Next, we characterize the structure of [Formula: see text] in which every non-nilpotent [Formula: see text]-invariant [Formula: see text]-subgroup is a TI-subgroup or a subnormal subgroup. Some previously known results are generalized.