Symmetric Generation and Existence of McL : 2, the Automorphism Group of the McLaughlin Group

Abstract

We use the primitive action of the Mathieu group M22 of degree 672 to define a free product of 672 copies of the cyclic group ℤ2 extended by M22 to form a semidirect product which we denote by P = 2☆672: M 22. Such a semidirect product is called a progenitor. By investigating a subprogenitor of shape 2☆42: A 7 we are led to a short relation by which to factor P. We verify that the resulting factor group is McL: 2, the automorphism group of the McLaughlin simple group, and identify it with the familiar permutation group of degree 275.