Timeline for How many conjugacy classes of subgroups does GL(2,p) have?
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7 events
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| May 27, 2010 at 20:45 | comment | added | Jack Schmidt | The classification seems quite poor if it cannot even produce the number of things it has classified. | |
| May 27, 2010 at 20:39 | comment | added | Robin Chapman | I'm not sure why you need the exact number of conjugacy classes of subgroups when one already has a good classification of them. For "diagonal subgroups" unless it only consists of scalar matrices, its normalizer will either consist of all the diagonal matrices, or all of them plus the "antidiagonal ones". So counting these will just be a matter of careful bookkeeping (which in this case I think life is too short for :-) ). The same goes for the remaining classes of subgroups. | |
| May 27, 2010 at 19:36 | comment | added | Jack Schmidt | Oh, and I guess it might be useful to mention that the diagonal case appears quite feasible. I believe one need only count how many subgroups Z/p^kZ x Z/p^kZ are invariant under the coordinate switch. I believe this function is probably a "polynomial" suitably interpreted. This produces a compact and easy to evaluate multiplicative formula for the real function. My methods of intuiting the correct formula and verifying correctness are quite slow, but I believe they are standard fare for anyone who understood combinatorics at some point in their life. | |
| May 27, 2010 at 19:29 | comment | added | Jack Schmidt | My question really is asking for the formula, not just feasibility. Perhaps just handling one of the special cases might be helpful. For instance, how many conjugacy classes of diagonal subgroups are there? That is, how many conjugacy classes of subgroups of GL(2,p) project onto the type "p-1" cyclic subgroups of PGL(2,p)? I am slowly piecing together this function, but my current methods are slow going, and there are still lots of cases left. Ideally the formula actually describes conjugacy class reps, but I can make do with some bounded searches as long as I have exact counts. | |
| May 27, 2010 at 14:09 | comment | added | Jim Humphreys |
As Robin says, it does seem feasible to arrive at some sort of count in this very special case. But for parallel results on finite groups of Lie type in general, starting with study of their subgroup structure, a lot more theory would have to be brought in from the algebraic group viewpoint. Aside from that, I'm unsure how much insight into a finite group of Lie type one can get by counting the number of conjugacy classes of subgroups. For larger groups than GL$(2,p)$ this would get arbitrarily complicated.
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| May 27, 2010 at 6:37 | history | edited | Robin Chapman | CC BY-SA 2.5 |
made more concise
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| May 27, 2010 at 5:57 | history | answered | Robin Chapman | CC BY-SA 2.5 |