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Classification Theorem of Finite Groups

The classification theorem of finite simple groups, also known as the "enormous theorem," which states that the finite simple groups can be classified completely into

1. Cyclic groups Z_p of prime group order,

2. Alternating groups A_n of degree at least five,

3. Lie-type Chevalley groups given by PSL(n,q), PSU(n,q), PsP(2n,q), and POmega^epsilon(n,q),

4. Lie-type (twisted Chevalley groups or the Tits group) ^3D_4(q), E_6(q), E_7(q), E_8(q), F_4(q), ^2F_4(2^n)^', G_2(q), ^2G_2(3^n), ^2B(2^n),

5. Sporadic groups M_(11), M_(12), M_(22), M_(23), M_(24), J_2=HJ, Suz, HS, McL, Co_3, Co_2, Co_1, He, Fi_(22), Fi_(23), Fi_(24)^', HN, Th, B, M, J_1, O'N, J_3, Ly, Ru, J_4.

The "proof" of this theorem is spread throughout the mathematical literature and is estimated to be approximately 15000 pages in length.

See also

Finite Group, Group, j-Function, Simple Group

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References

Cartwright, M. "Ten Thousand Pages to Prove Simplicity." New Scientist 109, 26-30, 1985.Cipra, B. "Are Group Theorists Simpleminded?" What's Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 82-99, 1996.Cipra, B. "Slimming an Outsized Theorem." Science 267, 794-795, 1995.Gorenstein, D. "The Enormous Theorem." Sci. Amer. 253, 104-115, Dec. 1985.Solomon, R. "On Finite Simple Groups and Their Classification." Not. Amer. Math. Soc. 42, 231-239, 1995.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 57, 1986.

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Classification Theorem of Finite Groups

Cite this as:

Weisstein, Eric W. "Classification Theorem of Finite Groups." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClassificationTheoremofFiniteGroups.html

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