Michael Vaughan-Lee - Oxford University Mathematical Institute

‎We investigate Graham Higman's paper Enumerating p-groups‎, ‎II‎, ‎in which he formulated his famous PORC conjecture‎. ‎We are able to simplify some of the theory‎. ‎In particular‎, ‎Higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎monomial equations is PORC‎. ‎It turns out that the homological algebra‎ ‎is just razzle dazzle‎, ‎and can all be replaced by the single observation‎ ‎that if you write the equations as the rows of a matrix then the number of‎ ‎solutions is the product of the elementary divisors in the Smith normal form‎ ‎of the matrix‎. ‎We obtain the PORC formulae for the number of r-generator groups of p‎ -‎class two for r\leq ۶‎. ‎In addition‎, ‎we obtain the PORC formula for the‎ ‎number of p-class two groups of order p^{۸}‎.

Enumerating p-groups, PORC conjecture, Graham Higman