The author introduces a generalization of both “being complemented” and his concept of -normality [J. Algebra 180, No. 3, 954-965 (1996; Zbl 0847.20010)] as follows: a subgroup of a group is said to be -supplemented (in ) if there exists a subgroup of such that and , the largest normal subgroup of contained in .
Theorem 3.1: Let be a finite group and let be a Sylow -subgroup of where is a prime divisor of with . Suppose that every maximal subgroup of is -supplemented in and any two complements of in are conjugate in . Then is -nilpotent and every -subgroup of is contained in some Hall -subgroup of . Theorem 3.3: Let be a finite group and let be a normal subgroup of such that is supersoluble. If every maximal subgroup of every Sylow subgroup of is -supplemented in , then is supersoluble. Theorem 4.2: Let be a finite group and let be the smallest prime divisor of . If is -free and every second-maximal subgroup of a Sylow -subgroup of is -normal in , then is -nilpotent. The last two theorems generalize results by A. Ballester-Bolinches and X. Guo [Arch. Math. 72, No. 3, 161-166 (1999; Zbl 0929.20015)].