Submission date: 30 July 2017

Abstract:

In two important papers from 1960 Graham Higman introduced the notion of PORC functions, and he proved that for any given positive integer n the number of p-class two groups of order p^n is a PORC function of p. A key result in his proof of this theorem is the following: “The number of ways of choosing a finite number of elements from the finite field of order q^n subject to a finite number of monomial equations and inequalities between them and their conjugates over GF(q), considered as a function of q, is PORC.” Higman's proof of this result involves five pages of homological algebra. Here we give a short elementary proof of the result. Our proof is constructive, and gives an algorithm for computing the relevant PORC functions.

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Full text arXiv 1707.09652: pdf, ps.