The previous post defined the groups PSL(*n*, *q*) where *n* is a positive integer and *q* is a prime power. These are finite simple groups for *n* ≥ 2 except for PSL(2, 2) and PSL(2, 3).

## Overlap among PSL(*n*, *q*)

There are a couple instances where different values of *n* and *q* lead to isomorphic groups: PSL(2, 4) and PSL(2, 5) are isomorphic, and PSL(2, 7) and PSL(3, 2) are isomorphic. These are the only instances [1].

With the exceptions stated above, distinct values of *n* and *q* lead to distinct groups. Is it possible for different choices of *n* and *q* to lead to groups of the same size, even though the groups are not isomorphic to each other? Yes, PSL(3, 4) and PSL(4, 2) both have order 20160, but the groups are not isomorphic. This is the only example [2].

## Overlap between PSL and alternating groups

The first post in this series mentioned that for *n* ≥ 5, the alternating group *A*_{n}, the group of even permutations on a set of *n* elements, is a simple group. Three of the alternating groups are isomorphic to PSL groups:

- PSL(2, 4) = PSL(2, 5) =
*A*_{5} - PSL(2, 9) =
*A*_{6} - PSL(4, 2) =
*A*_{8}

Here “=” really means isomorphic. We mentioned PSL(4, 2) above. It has the same order as PSL(3, 4). This means that *A*_{8} and PSL(3, 4) have the same order but are not isomorphic.

I suspect that with a small number of exceptions, the order of a finite simple group determines the group. I haven’t proven that, but numerical exploration suggests its true. This page lists non-Abelian finite simple groups of order less than 10 billion, and there are only seven orders that correspond to more than 1 group, the largest example being order 25,920.

## One last overlap

There is only one other duplication in the lists of groups in the CFSG theorem, and that is PSU(4, 2) = PSp(4, 3). I haven’t written about these groups yet.

## Notes

[1] See The Finite Simple Groups by Robert A. Wilson

[2] In fact, aside from the groups mentioned in this post, the orders of all the finite simple groups are unique except for two non-isomorphic families that have orders: PΩ_{2n+1}(*q*) and PSp_{2n}(*q*) for *n* ≥ 3 and odd prime powers *q*. See discussion on Math Overflow.

Each one of these overlaps, usually called “exceptional isomorphisms”, has a wonderful story associated to it. It means there are two different ways of looking at the same thing, and thus surprising relationships between two worlds.