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.

Key Words:

• McLaughlin
• Sporadic group
• Symmetric generation

2000 Mathematics Subject Classification:

• 20D08
• 20F05

Notes

In fact a syntheme is preserved by a subgroup of S₆ isomorphic to PGL₂(5), but we only have even permutations available in M₂₂.

Communicated by D. Macpherson.