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As a result of this previous question, I made the following video:

Cayley Tables of All Groups of Order 128, and what is striking is that most of them, if you squint, kind of resemble the elementary abelian group (gap, SmallGroups(128,2328)):

  • Is there a group-theoretical reason that this might be the case?
  • Is there a measure of how far one of these groups is away from being the elementary abelian group?
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You might start by explaining what exactly we are looking at. And then maybe try to give some hint of what you might mean by "resemble". – Marc van Leeuwen Mar 24 '13 at 6:37
We're looking at all the Cayley tables of groups from GAP. I chose the ordering from GAP, so we're looking at SmallGroup(128,n) for n from 1 to 2328. The coloring is linear so element 1 gets mapped to black and element 128 mapped to white. – deoxygerbe Mar 24 '13 at 6:46
It's clear what's being asked. OP sees a weird pattern in the Cayley table and he wants to know what's up with that. Perfectly fine question – Alexander Gruber Mar 29 '13 at 4:02

Firstly, cool movie!

I'm guessing here a bit since I don't know the precise details of how you created the movie and the precise ordering you chose (which might cluster things together in such a way as to 'hide' many cases that look differently). $128$ is very special and allows very little room for variation in the group structure. You will only see interesting subgroups of order $2,4,8, 16, 32, 64$ with the huge majority having subgroups of order $8$, $16$ (I believe). The elementary abelian group is not that far from a group where all elements have order at most, say $8$, so most groups will appear very similar to the elementary abelian group.

Again, this is mostly a guess. You might want to play with sizes that are not a multiple of just a single prime to see if that significantly alters the picture.

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"allows very little room for variation in the group structure" is totally wrong. Among small orders, those that are powers of two allow vastly more different group structures than other numbers of about the same size. See… – Marc van Leeuwen Mar 24 '13 at 6:39
I didn't mean little room for variation in group structure relative to group size, but rather relative to potential subgroup structures (not subgroup numbers). To clarify, a group of size 128 will (of course) only have subgroups possessing elements of order a power of 2 and every nontrivial subgroup will have an order 2 element. But a group of order 129 will have a subgroup order 3 or a subgroup of 7. In that respect, even though there are far less non-isomorphic groups of order 129 than there are of order 128, group size 129 allows more variation in subgroup structure than group size 128. – Ittay Weiss Mar 24 '13 at 6:46

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