Classification of Finite Simple Groups
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Theorem
The finite simple groups can be classified as follows:
Abelian Simple Groups
The Abelian simple groups are the cyclic groups of prime order.
Alternating Groups
The alternating groups $\map A n$ for $n \ge 5$.
Groups of Lie Type
The various families of groups of Lie type.
Sporadic Groups
The sporadic groups are as follows:
| Name | Abbreviation | Order | Date discovered | Discovered by |
|---|---|---|---|---|
| Mathieu Group | $\mathrm M_{11}$ | $2^4 \times 3^2 \times 5 \times 11$ | $1861$ | |
| Mathieu Group | $\mathrm M_{12}$ | $2^6 \times 3^3 \times 5 \times 11$ | $1861$ | |
| Mathieu Group | $\mathrm M_{22}$ | $2^7 \times 3^2 \times 5 \times 7 \times 11$ | $1873$ | |
| Mathieu Group | $\mathrm M_{23}$ | $2^7 \times 3^2 \times 5 \times 7 \times 11 \times 23$ | $1873$ | |
| Mathieu Group | $\mathrm M_{24}$ | $2^{10} \times 3^3 \times 5 \times 7 \times 11 \times 23$ | $1873$ | |
| Janko Group | $\mathrm J_1$ | $2^3 \times 3 \times 5 \times 7 \times 11 \times 19$ | $1966$ | |
| Janko Group | $\mathrm J_2$ | $2^7 \times 3^3 \times 5^2 \times 7$ | $1968$ | |
| Janko Group | $\mathrm J_3$ | $2^7 \times 3^5 \times 5 \times 17 \times 19$ | $1968$ | |
| Janko Group | $\mathrm J_4$ | $2^{21} \times 3^3 \times 5 \times 7 \times 11^3 \times 23 \times 29 \times 31 \times 37 \times 43$ | $1976$ | |
| Higman-Sims Group | $\mathrm {HS}$ | $2^9 \times 3^2 \times 5^3 \times 7 \times 11$ | $1968$ | |
| McLaughlin Group | $\mathrm {McL}$ | $2^7 \times 3^6 \times 5^3 \times 7 \times 11$ | $1969$ | |
| Suzuki Group | $\mathrm {Suz}$ | $2^{13} \times 3^7 \times 5^2 \times 7 \times 11 \times 13$ | $1969$ | |
| Rudvalis Group | $\mathrm {Ru}$ | $2^{14} \times 3^3 \times 5^3 \times 7 \times 13 \times 29$ | $1973$ | |
| Held Group | $\mathrm {He}$ | $2^{10} \times 3^2 \times 5^2 \times 7^3 \times 17$ | $1969$ | |
| Lyons Group | $\mathrm {Ly}$ | $2^8 \times 3^7 \times 5^6 \times 7 \times 11 \times 31 \times 37 \times 67$ | $1972$ | |
| O'Nan Group | $\text {O'N}$ | $2^9 \times 3^4 \times 5 \times 7^3 \times 11 \times 19 \times 31$ | $1976$ | |
| Conway Group | $\mathrm {Co3}$ | $2^{10} \times 3^7 \times 5^3 \times 7 \times 11 \times 23$ | $1969$ | |
| Conway Group | $\mathrm {Co2}$ | $2^{18} \times 3^6 \times 5^3 \times 7 \times 11 \times 23$ | $1969$ | |
| Conway Group | $\mathrm {Co1}$ | $2^{21} \times 3^9 \times 5^4 \times 7^2 \times 11 \times 13 \times 23$ | $1969$ | |
| Fischer Group | $\map {\mathrm {Fi} } {22}$ | $2^{17} \times 3^9 \times 5^2 \times 7 \times 11 \times 13$ | $1971$ | |
| Fischer Group | $\map {\mathrm {Fi} } {23}$ | $2^{18} \times 3^{13} \times 5^2 \times 7 \times 11 \times 13 \times 17 \times 23$ | $1971$ | |
| Fischer Group | ${\map {\mathrm {Fi} } {24} }'$ | $2^{21} \times 3^{16} \times 5^2 \times 7^3 \times 11 \times 13 \times 17 \times 23 \times 29$ | $1971$ | |
| Harada-Norton Group | $\mathrm {HN}$ | $2^{14} \times 3^6 \times 5^6 \times 7 \times 11 \times 19$ | $1976$ | |
| Thompson Group | $\mathrm {Th}$ | $2^{15} \times 3^{10} \times 5^3 \times 7^2 \times 13 \times 19 \times 31$ | $1976$ | |
| Baby Monster | $\mathrm {BM}$ | $2^{41} \times 3^{13} \times 5^6 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 31 \times 47$ | $1976$ | |
| Fischer-Griess Monster | $\mathrm {M}$ | $2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71$ | $1981$ |
Historical Note
The complete proof of the classification of the finite simple groups is spread over thousands of pages, published over the course of many years in several different journals.
It is a challenging exercise for a mathematician to check the entire proof for correctness and completeness.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): normal subgroup
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normal subgroup