# There are at most two finite simple groups of any order

From Groupprops

## Contents

## Statement

Let be a natural number. Then, there are at most two finite simple groups of order .

The smallest value of for which there are two non-isomorphic simple groups of order is . The two groups in this case are the alternating group of degree eight (which is also isomorphic to the projective special linear group ) and projective special linear group:PSL(3,4).

## Description of relevant pairs

### Infinite families

Note that there are examples other than the infinite families given below.

Below, is a prime power denoting the size of the field over which we are considering stuff.

Condition on | Condition on | First family (Chevalley notation) | First family (description) | Second family (Chevalley notation) | Second family (description) | Order |
---|---|---|---|---|---|---|

odd (when is even, the corresponding groups are in fact isomorphic) | (for or , the corresponding groups are in fact isomorphic) | Chevalley group of type B, more explicitly this is , the intersection of kernels of the Dickson invariant and spinor norm in the orthogonal group | projective symplectic group |

### First few examples

These include both the infinite families and the examples not arising from infinite families.

Order | First group | Second group |
---|---|---|

20160 | alternating group:A8 (isomorphic to ) | projective special linear group:PSL(3,4) |

4585351680 | Chevalley group of type B:B3(3) | projective symplectic group:PSp(6,3) |

## Proof

All known proofs of this fact employ the classification of finite simple groups, along with some explicit computations.