On the focal subgroup of a saturated fusion system

The influence of the cyclic subgroups of order p   or 4 of the focal subgroup of a saturated fusion system FF over a p-group S is investigated in this paper. Some criteria for normality of S   in FF as well as necessary and sufficient conditions for nilpotency of FF are given. The resistance of a p-group in which every cyclic subgroup of order p or 4 is normal, and earlier results about p-nilpotence of finite groups and nilpotency of saturated fusion systems are consequences of our study.