# Conjugacy of \(\mathrm {Alt}_5\) and \(\mathrm {SL}(2, 5)\) Subgroups of \(E_8(\mathbb C)\)

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*Darrin D. Frey*

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.

#### Table of Contents

# Table of Contents

## Conjugacy of $\mathrm{Alt}_{5}$ and $\mathrm{SL}(2, 5)$ Subgroups of $E_{8}(\mathbb C)$

- Table of Contents vii8 free
- Introduction and Preliminaries 110 free
- The Dihedral group of order 6 2635 free
- The Dihedral Group of order 10 3342
- The Alt[sub(5)] and SL(2,5) fusion patterns in G, A, Δ and Ω 4554
- Fusion patterns of Alt[sub(5)] and SL(2, 5) subgroups of H 7382
- Fusion patterns of Alt[sub(5)] subgroups of E 111120
- Conjugacy classes of Alt[sub(5)] subgroups of G 115124
- Conjugacy classes of SL(2,5) subgroups of G 135144
- Appendix 155164
- Table of Notation and Definitions 160169
- References 161170