There are at most two finite simple groups of any order

Statement

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

The smallest value of $n$ for which there are two non-isomorphic simple groups of order $n$ is $20160=2^{6}\cdot 3^{2}\cdot 5\cdot 7$. The two groups in this case are the alternating group of degree eight (which is also isomorphic to the projective special linear group $PSL(4,2)$) 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, $q=p^{r}$ is a prime power denoting the size of the field over which we are considering stuff.

Condition on $q$	Condition on $n$	First family (Chevalley notation)	First family (description)	Second family (Chevalley notation)	Second family (description)	Order
odd (when $q$ is even, the corresponding groups are in fact isomorphic)	$n>2$ (for $n=1$ or $n=2$, the corresponding groups are in fact isomorphic)	$B_{n}(q)$	Chevalley group of type B, more explicitly this is $\Omega _{2n+1}(q)$, the intersection of kernels of the Dickson invariant and spinor norm in the orthogonal group $O(2n+1,q)$	$C_{n}(q)$	projective symplectic group $PSp(2n,q)$	$\!q^{n^{2}}\left[\prod _{i=1}^{n}(q^{2i}-1)\right]/\operatorname {gcd} (2,q-1)$

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 $PSL(4,2)$)	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.

See also

• Non-isomorphic simple groups may have the same order

External links

See Oeis:A119648 for the list of numbers for which there are two non-isomorphic simple groups of said order.