Are there/Why aren't there any simple groups with orders like this?

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?

Solution 1:

I am posting the following counterexample to the question, as requested by caveman in the comments.

The Steinberg group ${}^2A_5(79^2)$ has order $$ 2^{23}\cdot 3^4\cdot 5^6\cdot 7^2\cdot 11^1\cdot 13^3\cdot 43^1\cdot 79^{15}\cdot 641^1\cdot 1091^1\cdot 3121^1\cdot 6163^2.$$

There are other counterexamples, too. For example ${}^2A_9(47^2)$ has order $$ 2^{43}\cdot 3^{13}\cdot 5^2\cdot 7^3\cdot 11^1\cdot 13^2\cdot 17^2\cdot 23^5\cdot 31^1\cdot 37^1\cdot 47^{45}\cdot 61^1\cdot 97^1\cdot 103^3\cdot 3691^1\cdot 5881^1\cdot 14621^1\cdot 25153^1\cdot 973459^1\cdot 1794703^1\cdot 4778021^2.$$

I would guess there are infinite counterexamples, but the numbers (of course) get very very large!

Solution 2:

Emil Artin showed in his 1955 papers$^1{}^2$ that groups of Lie type of characteristic $p$ have a large Sylow $p$-subgroup. That explains why the families $^2D_n,^2E_6,F_4,^2G_2$ and $E_n$ ($n=6,7,8$) have a "spike" at exactly their characteristic, and why the $2$-power in the orders of other groups of Lie Type of characteristic $2$ is much higher. It doesn't explain the underlying exponential pattern.

Orders of the alternating groups are easy to explain, so that leaves the sporadics. Of these, everything adheres to your observations except for the O'Nan group, which has has a little bump at $7$ (the order is $2^9 \cdot 3^4 \cdot 5 \cdot 7^3 \cdot 11 \cdot 19 \cdot 31$) and the same for $J_{11}$ (the order is $2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43$). These don't violate your predictions by very much, though, so I suspect the extra $7$'s and $11$'s are just a number theoretic coincidence.

We observe that in Sylow subgroups of other groups of Lie type, the characteristic is $2$, so their order has a peak at the $2$-power. It makes sense that the orders of these groups look like especially sharp exponential decay. Several of the sporadic groups (sporadics of characteristic type $2$, for which $F^\star(Y)$ is a $2$-group for every $2$-local subgroup $Y$) have been described$^3$ to me as sporadics "trying to have characteristic $2$." In particular, they have large Sylow $2$-subgroups just like like groups of Lie type of characteristic $2$, so they also have a peak at $2$. The rest of the sporadic groups, I have no intuitive explanation.

1. Emil Artin. "The orders of the classical simple groups." Communications in Pure and Applied Mathematics 8 (1955), 455–472.

2. Emil Artin. "The orders of the linear groups." Communications in Pure and Applied Mathematics 8 (1955). 355–365.

3. Thanks to G. Glauberman for this phrasing and his conversation on this topic.