On Graham Higman's famous PORC paper

Document Type: Research Paper

Author

Oxford University Mathematical Institute

Abstract

‎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 $r\leq 6$‎. ‎In addition‎, ‎we obtain the PORC formula for the‎ ‎number of $p$-class two groups of order $p^{8}$‎.

Keywords

Main Subjects


S. R. Blackburn, P. M. Neumann, and G.~Venkataraman (2007). Enumeration of finite groups. Cambridge Tracts in Mathematics, {bf 173}, Cambridge University Press, Cambridge.
W.~Bosma, J.~Cannon and C.~Playoust (1997). The {M}agma algebra system {I}: The user language. J. Symbolic Comput.. 24, 235-265
M. du~Sautoy and M. Vaughan-Lee (2012). Non-{PORC} behaviour of a class of descendant $p$-groups. J. Algebra. 361, 287-312
B. Eick and E. A. O'Brien (1999). Enumerating $p$-groups. J. Austral. Math. Soc. Ser. A. 67, 191-205
J. A. Green (1955). The characters of the finite general linear groups. Trans. Amer. Math. Soc.. 80, 402-447
G.~Higman (1960). Enumerating $p$-groups. {I}: Inequalities. Proc. London Math. Soc. (3). 10, 24-30
G.~Higman, (1960). Enumerating $p$-groups. {II}: Problems whose solution is {PORC}. Proc. London Math. Soc. (3). 10, 566-582
I. G. Macdonald (1995). Symmetric functions and {H}all polynomials. Second edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York.
M. F.~Newman, E. A.~O'Brien and M. R. Vaughan-Lee (2004). Groups and nilpotent {Lie} rings whose order is the sixth power of a prime. J. Algebra. 278, 383-401
L.~Pyber (1993). Enumerating finite groups of given order. Ann. of Math. (2). 137, 203-220
Charles~C. Sims (1965). Enumerating $p$-groups. Proc. London Math. Soc. (3). 15, 151-166
B. Witty (2006). Enumeration of groups of prime-power order. PhD thesis, Australian National University, http://www.brettwitty.net/phd.php.