TITLE: How many finite simple groups of order $p+1$?
QUESTION [6 upvotes]: I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number. 
But they don't seem to fall into any classification - have these all been determined? Is the number of them even finite?

REPLY [5 votes]: Standard heuristics (together with orders from the list of finite simple groups ) suggest that by far the most common orders of the form $p+1$ for $p$ prime will come from $A_1(q) = PSL_2(\mathbb{F}_q)$, of order $\frac{q^3-q}{2}$, as $q$ ranges over odd primes (or prime powers, if you want an additional small contribution).  In particular, for large $N$, one should expect roughly $\frac{\sqrt[3]{N}}{(\log N)^\alpha}$ satisfactory numbers $p+1$ less than $N$, for some fixed positive number $\alpha$, and this sequence of numbers certainly grows without bound.
As others have remarked, the question of proving that the set of suitable primes satisfies the rough asymptotics I gave above, or even proving that the set is infinite, seems to be beyond current technology.  For example, we still don't know if there are infinitely many primes of the form $n^2+1$ for $n$ an integer.