Cyclic extensions of finite simple groups

Abstract

Abstract Here cyclic extensions, not necessarily split, of simple groups are looked at. It is shown that if N is a finite simple group that is either an abelian group, an alternating group, a Suzuki group, a projective special linear group or a sporadic simple group and G = 〈 N , u 〉 ${G=\langle N,u\rangle}$ is a cyclic extension of N resulting in a Moufang loop, then 〈 N , u 2 〉 ${\langle N,u^{2}\rangle}$ is a group. Moreover, if G is nonassociative, then G is a generalization of the Chein loop where u inverts all of the elements of N.