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# Classification of finite simple groups

The Classification of finite simple groups consists of a large number of articles written by over a hundred authors, mostly published between the 1960s and 1983, in which year Daniel Gorenstein announced that all finite simple groups were finally classified. The total number of pages of the articles required for the completion of the proof is estimated to be over 15000; a simplified and revised version of the proof is being worked on.

The theorem states that every finite simple group is isomorphic to a group of (at least) one of the following four classes of groups:

- Cyclic groups of prime order
- Alternating groups of degree
- Simple groups of Lie type
- The 26 sporadic simple groups (of which the 2 groups with the largest orders are called the Baby Monster group and the Monster group)

## Sequences

The orders of cyclic groups of prime order are the primes (OEIS: A000040):

- {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, ...}

The orders of the simple alternating groups are half the factorial value of their respective degree (OEIS: A119631):

- {60, 360, 2520, 7920, 20160, 95040, 175560, 181440, 443520, 604800, 1814400, 9999360, 10200960, 13685760, 17971200, 19958400, 44352000, 50232960, 174182400, 197406720, 211341312, 239500800, 244823040, 898128000, ...}

The orders of the sporadic simple groups are (OEIS: A001228):

- {7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, 244823040, 898128000, 4030387200, 145926144000, 448345497600, 460815505920, 495766656000, 42305421312000, 64561751654400, 273030912000000, 51765179004000000, 90745943887872000, 4089470473293004800, 4157776806543360000, 86775571046077562880, 1255205709190661721292800, 4154781481226426191177580544000000, 808017424794512875886459904961710757005754368000000000}

## See also

- OEIS: A000040 Prime numbers.
- OEIS: A119631 Orders of simple alternating groups.
- OEIS: A001228 Orders of sporadic simple groups.