This is a preliminary discussion of my knowledge about 2-groups. There are still many questions I have, and I would appreciate any insights.
When we first learn about group theory, we learn some theorems about how to decompose finite groups, most notably the Sylow theorems, which tell us (roughly) how to decompose groups based on their maximal -power subgroups. So, we might expect that for random finite groups, we can obtain a lot of information about them by analyzing what happens at each prime factor of the order.
And indeed, groups that appear “in nature” tend to have several prime factors, and their Sylow subgroups are typically interesting objects in their own right. However, if we count groups not by their occurrences in nature but simply by their orders, then we see something very different: most finite groups are 2-groups: their order is a power of 2, so there is no information to be obtained by looking at other primes.
I was surprised at first to learn that the vast majority of groups are 2-groups: for example, there are 49,910,529,484 group of order up to 2000, and of those, 49,487,365,422, or more than 99%, have order . And the proportion of groups with 2-power order will only increase to 1 as we go further.
I don’t really understand why this is true (I have only barely skimmed the relevant papers, such as Eick and O’Brien’s “Enumerating -groups” and Besche, Eick, and O’Brien’s “The groups of order at most 2000”), but here is my vague intuition. The Jordan-Hölder theorem tells us that we can assemble groups out of simple groups. Now, it is likely possible to glue two large simple groups together in quite a few ways, whereas there are fewer ways of assembling two small simple groups together. However, more relevant than the size of the pieces is the number of pieces: a Jordan-Hölder series for a group of order 1024 will contain ten factors, each isomorphic to . As we all learned when we were very young and trying to assemble jigsaw puzzles, there are lots of ways of putting together many small pieces, but there are not so many ways of putting together a few large pieces. And this is just as true of groups as it is of jigsaw puzzles.
But there’s a problem with this analysis: there are way more groups with order 1024 than there are with , even though they contain the same number of factors in their composition series. In fact, there are 408,641,062 of order 1536, which is only roughly 40 times as many as there are groups of order (as opposed to nearly 5000 times for 1024). I would like to understand this discrepancy, but I’m not there yet.
One obstacle to understanding this point for me is that there isn’t such a discrepancy for groups of order versus when is small: for example, when and , there are 14 groups of order 16, 15 of order 24, 14 of order 40, 13 of order 56, and so forth. Similarly, when , there are 51 groups of order 32, 52 of order 48, 52 of order 80, and 43 of order 112. Since I don’t know any exciting examples of groups of order for large, it’s hard for me to gain intuition about what’s going on here.
There are lots of things to look into in the future, such as reading the papers by Besche and Eick. Perhaps I’ll report on them in the future.