Rank one isolated p-minimal subgroups in finite groups

This paper studies, for p a prime, rank one isolated p-minimal subgroups P in a finite group G. Such subgroups share many of the features of the minimal parabolic subgroups in groups of Lie type. The structure of Y, the normal closure in G of O ^p(P) is determined where O ^p(P) is the smallest normal subgroup of P such that P/O ^p(P) is a p-group. We find that if Y≠O ^p(P) and O _p(G)=1, then either Y/Z(Y) is a simple group of Lie type in characteristic p or p≤7 with Y/Z(Y) given by an explicit list. Of particular note is that twenty four out of the twenty six sporadic simple groups arise as possibilities for Y/Z(Y). This may be viewed as giving an overarching framework which brings together the simple groups of Lie type and (most of) the sporadic simple groups.