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Conjugacy of \(\mathrm{Alt}_5\) and \(\mathrm{SL}(2, 5)\) Subgroups of \(E_8(\mathbb C)\)
Darrin D. Frey, Winona State University, MN

Memoirs of the American Mathematical Society
1998; 162 pp; softcover
Volume: 133
ISBN-10: 0-8218-0778-1
ISBN-13: 978-0-8218-0778-1
List Price: US$54
Individual Members: US$32.40
Institutional Members: US$43.20
Order Code: MEMO/133/634
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Exceptional complex Lie groups have become increasingly important in various fields of mathematics and physics. As a result, there has been interest in expanding the representation theory of finite groups to include embeddings into the exceptional Lie groups. Cohen, Griess, Lisser, Ryba, Serre and Wales have pioneered this area, classifying the finite simple and quasisimple subgroups that embed in the exceptional complex Lie groups.

This work contains the first major results concerning conjugacy classes of embeddings of finite subgroups of an exceptional complex Lie group in which there are large numbers of classes. The approach developed in this work is character theoretic, taking advantage of the classical subgroups of \(E_8 (\mathbb C)\). The machinery used is relatively elementary and has been used by the author and others to solve other conjugacy problems. The results presented here are very explicit. Each known conjugacy class is listed by its fusion pattern with an explicit character afforded by an embedding in that class.


Graduate students and research mathematicians interested in \(E_8 (\mathbb C)\); physicists working in string theory or quantum mechanics.

Table of Contents

  • Introduction and Preliminaries
  • The dihedral group of order 6
  • The dihedral group of order 10
  • The \(\mathrm{Alt}_5\) and \(\mathrm{SL}(2, 5)\) fusion patterns in \(\mathrm{G}, \mathcal A, \Delta\) and \(\Omega\)
  • Fusion patterns of \(\mathrm{Alt}_5\) and \(\mathrm{SL}(2, 5)\) subgroups of \(\mathrm{H}\)
  • Fusion patterns of \(\mathrm{Alt}_5\) subgroups of \(\mathcal E\)
  • Conjugacy classes of \(\mathrm{Alt}_5\) subgroups of \(\mathrm{G}\)
  • Conjugacy classes of \(\mathrm{SL}(2, 5)\) subgroups of \(\mathrm{G}\)
  • Appendix
  • Table of notation
  • References
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