Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 009, 18 pages      arXiv:1101.3055      http://dx.doi.org/10.3842/SIGMA.2011.009
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Introduction to Sporadic Groups

Luis J. Boya
Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received September 18, 2010, in final form January 12, 2011; Published online January 16, 2011

Abstract
This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the 1+1+16=18 families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated ''pariah'' groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group M, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the 5+7+8+6=26 sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.

Key words: group theory; finite groups.

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