There are at most two finite simple groups of any order
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
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)|
All known proofs of this fact employ the classification of finite simple groups, along with some explicit computations.