Cyclic extensions of finite simple groups (Q524634)

It is of natural interest to look at cyclic extensions, not necessarily split, of simple groups and to determine when a Moufang loop, which is obtained using an upward extension of a simple group by a cyclic group, is guaranteed to be a group. Suppose that \(G\) is a finite Moufang loop with \(G=NH\) where \(H=\langle u \rangle\) and \(N \trianglelefteq G\) is a simple group. If \(N\) is an abelian group, a Suzuki group, an alternating group, a projective special linear group or a sporadic simple group then \(\langle N, u^2\rangle\) is a group. Moreover, if \(G\) is nonassociative, then \(G=N\rtimes H\) is a generalization of the Chein loop where \(u\) inverts all of the elements of \(N\).