Fred, another interesting question: what are the maximal subgroups of PGL_2(GF(p)) (say for p an odd prime)? Hint: besides the Cartan and Borel subgroups, there are a finite collection which is well-known in a different context. Victor On Wed, Mar 20, 2013 at 9:51 PM, Victor S. Miller <victorsmiller@gmail.com>wrote:
Fred, Let F=GF(p^2) as a vector space over GF(p). Let sigma be the matrix which raises an element of F to the p-th power. Every 2 by 2 matrix can be written uniquely as a*sigma + b, where a and b are elements of F. In modern terms you want to show that every element in GL_2 is in a Cartan subgroup -- a conjugate of multiplication by an element of F and its powers, or in a Borel subgroup, which is conjugate to the subgroup of powers of F.
Victor
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On Mar 20, 2013, at 21:19, Fred lunnon <fred.lunnon@gmail.com> wrote:
The following kept me at bay for an embarrassing length of time, despite my conviction that there just had to be a simple, one --- well, maybe three --- line proof (there is).
Question (elementary): for p odd prime, show that the order of any element of PGL(2, p) divides some member of {p-1, p, p+1} .
Rider (harder): locate where the @£$%^&* this might be concealed amidst the 330-odd densely obfuscated pages of L. E. Dickson (1901, 1958) "Linear groups with an exposition of the Galois field theory"
Some things were quite definitely not better in the old days!
Fred Lunnon
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