Classification Theorem

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 $\Bbb{Z}_p$ of Prime Order,

2. Alternating Groups $A_n$ of degree at least five,

3. Lie-Type Chevalley Groups ${\it PSL}(n,q)$, ${\it PSU}(n,q)$, ${\it PsP}(2n,q)$, and $P\Omega^\epsilon(n,q)$,

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 $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $J_2={\it HJ}$, Suz, HS, McL, ${\it Co}_{3}$, ${\it Co}_{2}$, ${\it Co}_{1}$, He, ${\it Fi}_{22}$, ${\it Fi}_{23}$, ${\it Fi}'_{24}$, 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 15,000 pages in length.

See also Finite Group, Group, j-Function, Simple Group

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.