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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A correlation between $ {\rm PSU}\sb{4}\,(3)$, the Suzuki group, and the Conway group


Author: J. H. Lindsey
Journal: Trans. Amer. Math. Soc. 157 (1971), 189-204
MSC: Primary 20.75
MathSciNet review: 0283097
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Abstract: We shall use a six dimensional projective representation of $ PS{U_4}(3)$ of order $ {2^7}{3^6}5 \cdot 7$ to construct 12 and $ 24$-dimensional complex projective representations of the Suzuki and Conway groups, respectively, acting on the Leech lattice. The construction makes it easy to show that the Suzuki and Conway simple groups have outer automorphism groups of order two and one, respectively. Also, the simple Suzuki group contains $ 3 \cdot PS{U_4}(3) \cdot 2,{3^5} \cdot {M_{11}}$, and a group which is probably $ PS{U_5}(2)$, where $ A \cdot B$ denotes an extension of the group $ A$ by the group $ B$.


References [Enhancements On Off] (What's this?)

  • [1] J. H. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969), 79–88. MR 0248216 (40 #1470)
  • [2] Richard Brauer, Über endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73–96 (German). MR 0206088 (34 #5913)
  • [3] J. H. Lindsey II, Linear groups with an irreducible, normal, rank two $ p$-subgroup (to appear).
  • [4] -, Finite linear groups of degree six (to appear).
  • [5] -, On a six-dimensional projective representation of $ PS{U_4}(3)$, Pacific J. Math. 36 (1971).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0283097-8
PII: S 0002-9947(1971)0283097-8
Keywords: $ PS{U_4}(3)$, Suzuki group, Conway group, Leech lattice, orbit, automorphism group, centralizer of a central involution
Article copyright: © Copyright 1971 American Mathematical Society