The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this:

```
2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71
```

starts with a very high power of 2, then the powers decrease and you get a tail - it's something like exponential decay.

Why does this happen? I want to understand this phenomenon better.

I wanted to find counter-examples, e.g. a simple group of order something like

```
2^4 3^2 11^5 13^9
```

but it seems like they do not exist (unless it slipped past me!).

We have the following bound $|G| \le \left(\frac{|G|}{p^k}\right)!$ which allows $3^2 11^4$ but rules out orders like $3^2 11^5$, $3^2 11^6$, .. while this does give a finite bound it is extremely weak when you have more than two primes, it really doesn't explain the pattern but a much stronger bound of the same type might?

I also considered that it might be related to multiple transitivity, a group that is $t$-transitive has to have order a multiple of $t!$, and e.g. 20! =

```
2^18 3^8 5^4 7^2 11 13 17 19
```

which has exactly the same pattern, for reasons we do understand. But are these groups really transitive enough to explain the pattern?