# Classification of Finite Simple Groups

## 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 $A \left({n}\right)$ for $n \ge 5$.

### Groups of Lie Type

The various families of groups of Lie type.

The sporadic finite simple 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 $\mathrm J_4$ $2^{21} \times 3^3 \times 5 \times 7 \times 11^3 \times 23 \times 29 \times 31 \times 37 \times 43$ $1976$
$\mathrm {HS}$ $2^9 \times 3^2 \times 5^3 \times 7 \times 11$ $1968$
$\mathrm {McL}$ $2^7 \times 3^6 \times 5^3 \times 7 \times 11$ $1969$
$\mathrm {Suz}$ $2^{13} \times 3^7 \times 5^2 \times 7 \times 11 \times 13$ $1969$
$\mathrm {Ru}$ $2^{14} \times 3^3 \times 5^3 \times 7 \times 13 \times 29$ $1973$
$\mathrm {He}$ $2^{10} \times 3^2 \times 5^2 \times 7^3 \times 17$ $1969$
$\mathrm {Ly}$ $2^8 \times 3^7 \times 5^6 \times 7 \times 11 \times 31 \times 37 \times 67$ $1972$
$\text {O'N}$ $2^9 \times 3^4 \times 5 \times 7^3 \times 11 \times 19 \times 31$ $1976$
$\mathrm {Co3}$ $2^{10} \times 3^7 \times 5^3 \times 7 \times 11 \times 23$ $1969$
$\mathrm {Co2}$ $2^{18} \times 3^6 \times 5^3 \times 7 \times 11 \times 23$ $1969$
$\mathrm {Co1}$ $2^{21} \times 3^9 \times 5^4 \times 7^2 \times 11 \times 13 \times 23$ $1969$
$\mathrm {Fi} \left({22}\right)$ $2^{17} \times 3^9 \times 5^2 \times 7 \times 11 \times 13$ $1971$
$\mathrm {Fi} \left({23}\right)$ $2^{18} \times 3^{13} \times 5^2 \times 7 \times 11 \times 13 \times 17 \times 23$ $1971$
$\mathrm {Fi} \left({24}\right)'$ $2^{21} \times 3^{16} \times 5^2 \times 7^3 \times 11 \times 13 \times 17 \times 23 \times 29$ $1971$
$\mathrm {HN}$ $2^{14} \times 3^6 \times 5^6 \times 7 \times 11 \times 19$ $1976$
$\mathrm {Th}$ $2^{15} \times 3^{10} \times 5^3 \times 7^2 \times 13 \times 19 \times 31$ $1976$
$\mathrm {BM}$ $2^{41} \times 3^{13} \times 5^6 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 31 \times 47$ $1976$
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.