Is there some reason for there to be more groups with 16 conjugacy classes than with 15 or 17?

It is a well-known exercise to show that a group with one conjugacy class has only one element, a finite group with two conjugacy classes must be cyclic of order two, and a finite group with three conjugacy classes must be cyclic of order three or non-abelian of order six. At some point I (along with everyone and their brother) classified finite groups with four conjugacy classes, but I never really looked beyond that until today.

My results are only partial, but I couldn't help noticing local maximums in my census at 10 classes (gentle), 14 classes (medium), 16 classes (sharp), and 18 classes (medium), with corresponding dips at 11 (gentle), 15 (sharp), and 17 (sharp).

Off hand I can't think of why a group might be more likely to have an even number of classes than an odd number, but perhaps this is well known. Four and five classes are relatively well known, as in one of my favorite papers, Miller (1919).

  • Miller, G. A. "Groups possessing a small number of sets of conjugate operators." Trans. Amer. Math. Soc. 20 (1919), no. 3, 260–270. MR1501126 JFM47.0094.04 DOI:10.2307/1988867
  • $\begingroup$ Do you actually know how many groups there are with exactly 18 conjugacy classes? oeis.org/A073043 lists the number of groups with exactly $n$ conjugacy classes, but it only goes up to $n=12$. Anyway, I wonder whether the references cited there would shed any light on the question. $\endgroup$ – Gerry Myerson Jun 23 '11 at 1:05
  • $\begingroup$ A little websearching turns up A. Vera-López and Josu Sangroniz, The finite groups with thirteen and fourteen conjugacy classes, Math. Nachr. 280 (2007), no. 5-6, 676–694, MR2308490 (2008b:20036). $\endgroup$ – Gerry Myerson Jun 23 '11 at 1:32
  • $\begingroup$ The oeis entry is wrong for n=12, easily checked in gap and Vera-Lopez^2 (1986). I didn't see anything about the even-odd distinction in the refs. $\endgroup$ – Jack Schmidt Jun 23 '11 at 1:52

Your number 16 reminds me of the beautiful theorem that for a group G of odd order, k(G) $\equiv$ |G| (mod 16). Here k(G) is the number of conjugacy classes of G.

And I would like to add that in 1903 Edmund Landau proved that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes. However, his proof does not say anything about the frequencies with which conjugacy class numbers arise.

  • 1
    $\begingroup$ That is a theorem of W. Burnside, by the way. $\endgroup$ – Geoff Robinson Apr 28 '16 at 9:56

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