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?
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?
The philosophical point here is that if all you know about a group $G$ is its order $\lvert G \rvert$, then by far the most relevant information is the prime factorization of that order. (Back when sporadic groups were still being discovered, there are anecdotes about phoning John Thompson with the order of your hypothetical new group, and after some calculations he would tell you whether it 'checked out' or not - and of course he would just be using the knowledge of which primes divided the order and to which exponents). So questions about the prime factorization of $\lvert G \rvert - 1$ are going to be dominated by the (generally unsolved) number theoretical problems that relate the prime factorization of $n$ and $n+1$, e.g. the existence of infinitely many Sophie Germain primes, Mersenne primes, or primes $p$ for which $\frac{p^2+1}{2}$ is also prime, etc.
As suggested by the other answers and comments, this is unknown (and a hard arithmetic question). Here's another example that might help indicate why:
The order of the finite simple group $PSL_2(F_q)$ is (for $q$ not a power of $2$) $q(q^2-1)/2$. You'd therefore like to know when $q(q^2-1)/2-1$ is a prime, for $q$ a prime power. The question is (at least superficially, I hope we can agree) similar to that of when the Mersenne number $2^n-1$ is prime.
For $q=2^n$ a power of $2$ the question boils down to asking when $2^n(2^{2n}-1)-1$ is prime.
There are similar formulas for the other simple groups of Lie type, and I'll bet money no one in the world knows whether infinitely many of the relevant numbers are prime.
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.
As Alon remarks, it is extremely hard to find such groups even between groups of one (most known) series $A_n$.