``Moonshine'' of finite groups.
This book is based on the author's lecture notes for a two-quarter course at Ohio State University in 1983-84. The term ``moonshine'' means a mysterious relation between finite groups and modular functions. The moonshine for the Monster simple group, which is the original one, was studied by \textit{J. H. Conway} and \textit{S. P. Norton} [Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)]. Since then, a number of works have been done on the subject, see for example \textit{I. Frenkel, J. Lepowsky} and \textit{A. Meurman}, [Vertex operator algebras and the Monster. Pure and Applied Mathematics 134. Boston: Academic Press (1988; Zbl 0674.17001)] and \textit{T. Gannon}, [Moonshine beyond the Monster. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (2006; Zbl 1146.11026)]. Although the lecture notes were written more than 25 years ago, the contents of the present book are essentially unchanged and only the bibliography is updated. This is mainly because the author discusses very basic materials. The book consists of four chapters. The first two are preparatory ones and proofs for theorems mentioned there are usually omitted. Chapter 1, entitled ``Modular functions and modular forms,'' is devoted to standard descriptions of modular functions and modular forms. Chapter 2, entitled ``Dedekind eta function,'' deals with basic properties of the Dedekind eta function \(\eta(z)\). In Chapter 3, entitled ``Moonshine of finite groups,'' the author discusses the main ingredients including generalized partitions and frame shapes. Chapter 4, entitled ``Multiplicative product of eta function,'' reviews a conjecture of J. McKay on products of \(\eta(z)\). The author follows \textit{D. Dummit, H. Kisilevsky} and \textit{J. McKay} [Finite groups -- coming of age, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)] and \textit{M. Koike} [Nagoya Math. J. 95, 85-89 (1984; Zbl 0548.10018)] for its proof. The book serves as a gentle introduction to ``moonshine,'' a fascinating field of mathematics.
scientific article; zbMATH DE number 5796327