1,537 research outputs found
Graham Higman's PORC theorem
Graham Higman published two important papers in 1960. In the first of these
papers he proved that for any positive integer the number of groups of
order is bounded by a polynomial in , and he formulated his famous
PORC conjecture about the form of the function giving the number of
groups of order . In the second of these two papers he proved that the
function giving the number of -class two groups of order is PORC. He
established this result as a corollary to a very general result about vector
spaces acted on by the general linear group. This theorem takes over a page to
state, and is so general that it is hard to see what is going on. Higman's
proof of this general theorem contains several new ideas and is quite hard to
follow. However in the last few years several authors have developed and
implemented algorithms for computing Higman's PORC formulae in special cases of
his general theorem. These algorithms give perspective on what are the key
points in Higman's proof, and also simplify parts of the proof.
In this note I give a proof of Higman's general theorem written in the light
of these recent developments
The number of regular semisimple conjugacy classes in the finite classical groups
Using generating functions, we enumerate regular semisimple conjugacy classes
in the finite classical groups. For the general linear, unitary, and symplectic
groups this gives a different approach to known results; for the special
orthogonal groups the results are new.Comment: 19 page
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