I just ran across
FiniteGroupData and related functions in Mathematica. That would have made some of my earlier posts easier to write had I used this instead of writing my own code.
Here’s something I find interesting. For each n, look at the groups of order at most n and count how many are Abelian versus non-Abelian. At first, there are more Abelian groups, but the non-Abelian groups soon become more numerous. Also, the number of Abelian groups grows smoothly, while the number of non-Abelian groups has big jumps, particularly at powers of 2.
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).
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 .