The Mathieu Groups

The Mathieu groups have stirred the imaginations of mathematicians for a century and a half since their original discovery by Emilé Mathieu in 1861 and 1873.  To steal the opening quote from this book: “There are almost as many different constructions of \( M_{24} \) as there have been mathematicians interested in that most remarkable of all finite groups.” And indeed there are, ranging from transitive extensions of finite linear groups, to symmetric generation, automorphisms of the Golay Code or automorphisms of Steiner Systems, and even using dessin d’enfants.

 

A. A. Ivanov’s The Mathieu Groups tells the story of \( M_{24} \) from the viewpoint of group amalgams. Ivanov’s presentation falls under the more general heading of Ivanov-Schpectorov geometries. The text essentially consists of two parts. The first part, Chapters 1-5, is devoted to the construction of the largest of the Mathieu groups, \( M_{24} \), and aspects of related small linear groups. More specifically, Ivanov constructs \( M_{24} \) using the Mathieu amalgam, a certain constrained rank-3 amalgam.

 

In the second part, Chapters 6-10, Ivanov does a perfect job of illustrating the utility of group amalgams in the study of \( M_{24} \) and the other Mathieu groups. The proofs are quite beautiful in their simplicity: the smaller Mathieu groups are found using completions of subamalgams of the Mathieu amalgam, the 45-dimensional representations of \( M_{24} \) are constructed by using the representation theory of the 3 groups in the Mathieu amalgam, the Held group is constructed using a related amalgam, and some of the maximal subgroups of the Mathieu groups are also found.

 

The material covered in this book make it excellent for graduate students, researchers, or really anyone with a passing interest in \( M_{24} \). The text is superbly written, clear, concise and yet mostly self-contained. Anyone who has read any standard introduction to the theory of finite groups will be able to follow along as if they were reading a novel. For those new to \( M_{24} \), a graduate algebra course should be all you need so long as you have a standard reference book on finite groups.

 

To summarize, The Mathieu Groups, by A. A. Ivanov, is an excellent chapter in the Mathieu story. My only complaint is that I didn’t have this book on hand when I was learning about the “most remarkable of all finite groups”.

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Bronson Lim is a Research Assistant Professor at the University of Utah in Salt Lake City. His research areas include group theory, equivariant differential geometry, and equivarant algebraic geometry. His email is bronson@math.utah.edu.