I was trying to work out the Sylow p-subgroups for general linear groups over arbitrary fields, and was running into some trouble with non-algebraically closed fields. The real numbers, R, were particularly troubling:
Does the group GL(2,R) of 2×2 invertible matrices with real entries have a single conjugacy class of maximal 2-subgroups?
The specific thing that bothers me is that every 2nth root of unity has a quadratic minimal polynomial over R, and so we get cyclic 2-subgroups of arbitrary (finite) size, but I'm not sure I understand how to write down the elements of the union of this subgroup, and so I am not sure what its normalizer looks like.
Is there a description of this subgroup that allows one to determine if any particular matrix is contained in it?
In particular, I can write the subgroup down as a direct limit of reasonably explicit matrix groups, but the maps are not the standards inclusions, and so I cannot easily describe the limit as an actual union.
Using trig identities we can write down a recursive formula for the minimal polynomial of the 2nth root of unity over R, and then the rational canonical form gives a generator for a cyclic group of order 2n. A generator for 2n is mapped to the 2mth power of the 2n+m generator, which for m = 1 (the only map a sane person cares about) is easy to write down explicitly. However, as a whole the direct limit seems quite inexplicit, since no particular matrix is ever actually in the direct limit, only a conjugate of it by an infinite sequence of change of bases.
I would like to give fairly constructive descriptions of the Sylow subgroups, and in particular, would like an algorithm that, for an arbitrary p-subgroup, finds a conjugator into a "standard" copy of the Sylow p-subgroup. However, right now I cannot even find the standard copy.