The Art of Working with the Mathieu Group M24

The Leech lattice Λ, the Conway group ∙O, and the Monster group M are immensely famous structures. They each grow out of the Mathieu group M24 and its underlying combinatorial structure, and play an important role in various branches of mathematics and in theoretical physics. Written by an expert in the field, this book provides a new generation of mathematicians with the intimate knowledge of M24 needed to understand these beautiful objects, and many others. It starts by exploring Steiner systems, before introducing the Miracle Octad Generator (MOG) as a device for working with the Steiner system S(5,8,24). Emphasizing how theoretical and computational approaches complement one another, the author describes how familiarity with M24 leads to the concept of 'symmetric generation' of groups. The final chapter brings together the various strands of the book to produce a nested chain of groups culminating in the largest Conway simple group Co1.

• Assumes little more than undergraduate-level material to remain accessible to a wide mathematical audience
• Demonstrates the interaction between theoretical and computational approaches, and how the two complement and enhance one another
• Shows how familiar, well-understood ideas lead inexorably to large and highly complex structures, including demonstrating how looking at the classical unitary group U3(3) in a certain way leads to the largest Conway group