2, 3, 13, 19, 17, 11, 23, 5, 10, 14, 20, 21, 16, 22, 18, 15, 4, 6, 12, 9, 7, 8

The rather bizarre sequence in the title is the ordering of the integers from 2 to 23 which is induced by the values of the Chebotarev invariant of the group An. In other words, if n comes before m in this list (for instance, n=20, m=9), it means that on average, you will need to pick fewer conjugacy classes in An at random to obtain a set that necessarily generates it, than you need to do for Am.

This illustrates some non-obvious property of this invariant; for instance, it is easier to “fill up” A21 than A8, despite the fact that the first has more maximal subgroups (14 against 6) and many more conjugacy classes (408 against 14) — in fact, there’s probably some money to be made by setting up betting games based on this type of observations.

The actual values of the invariants are in fact fairly close (except for the degenerate cases 2 and 3), ranging from 4.01632 or so to 4.939. It is not at all clear (to me) what the limiting behavior will be as n grows; I had first a vague guess that the sequence of invariants would grow to infinity (maybe slowly), but the data is more consistant with a convergence (or an oscillating behavior), which is what D. Zywina suggested to me for the symmetric group — where the behavior is roughly similar but the ordering is different.

I have been able to do these computations thanks to having again access to Magma; and I once more can’t help expressing my admiration for the incredible work that this team has done. (Admiration and gratitude which also apply equally to the members of the other teams working on algebraic-computation packages such as GAP, Pari/GP and Sage, even though for this particular type of computations, Magma is currently far ahead).

Published by

Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

2 thoughts on “2, 3, 13, 19, 17, 11, 23, 5, 10, 14, 20, 21, 16, 22, 18, 15, 4, 6, 12, 9, 7, 8”

  1. My hunch (on convergence) was already conjectured by finite group theorists! You should look at section 3.3 of http://www.math.carleton.ca/~jdixon/Prgrpth.pdf
    There has been some progress, but it seems like the problem is still open (which I find sort of surprising).

    By the way, in your betting game, a good scheme would be to only wager on primes n.
    Your list does a good job separating primes and composites (except for the rogue 7). This is likely due to the fact that A_n has no “imprimitive” maximal subgroups if n is prime.

Leave a Reply

Your email address will not be published. Required fields are marked *