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Ree Group

A Ree group is a twisted Chevalley group constructed by combining a field automorphism with an exceptional symmetry of a root system that interchanges long and short roots in certain Lie algebras. The two families are ^2G_2(q) for q=3^(2n+1) and ^2F_4(q) for q=2^(2n+1), where n=0,1,2,....

Their group orders are

|^2G_2(q)|=q^3(q^3+1)(q-1)

and

|^2F_4(q)|=q^(12)(q^6+1)(q^4-1)(q^3+1)(q-1).

The groups in both families are simple apart from their smallest members. In particular, ^2G_2(3) is not simple, and the derived subgroup of ^2F_4(2) is the simple Tits group.

See also

Chevalley Groups, Simple Group, Suzuki Group, Tits Group, Twisted Chevalley Groups

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References

Carter, R. W. Simple Groups of Lie Type. New York: Wiley, 1989.Ree, R. "A Family of Simple Groups Associated with the Simple Lie Algebra of Type (G_2)." Bull. Amer. Math. Soc. 66, 508-510, 1960. https://doi.org/10.1090/S0002-9904-1960-10523-X.Ree, R. "A Family of Simple Groups Associated with the Simple Lie Algebra of Type (F_4)." Amer. J. Math. 83, 401-420, 1961. https://doi.org/10.2307/2372886.

Cite this as:

Weisstein, Eric W. "Ree Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReeGroup.html

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