On Graham Higman's famous PORC paper

• Main Author: Michael Vaughan-Lee
• Format: Article
• Language: English
• Published: University of Isfahan 2012-12-01
• Series: International Journal of Group Theory
• Subjects: Enumerating p-groups; PORC conjecture; Graham Higman
• Online Access: http://www.theoryofgroups.ir/?_action=showPDF&article=1591&_ob=6206fc955e4cba86c276b42fd3c4770f&fileName=full_text.pdf

We investigate Graham Higman's paper Enumerating p-groups, II, in whichhe formulated his famous PORC conjecture. We look at the possibilities forturning his theory into a practical algorithm for computing the number of p-class two groups of order pn for small n. We obtain the PORC formulae for thenumber of r-generator groups of p-class two for r 6. In addition, we obtainthe PORC formula for the number of p-class two groups of order p8.One of the ideas used in implementing Higman's theory has led to a signi -cant speed up in Eamonn O'Brien's ClassTwo function in Magma. In addition,we are able to simplify some of the theory. In particular, Higman's paper con-tains ve pages of homological algebra which he uses in his proof that thenumber of solutions in a nite eld to a nite set of monomial equations isPORC. It turns out that the homological algebra is just razzle dazzle, and canall be replaced by the single observation that if you write the equations as therows of a matrix then the number of solutions is the product of the elementarydivisors in the Smith normal form of the matrix.