Graham Higman's PORC theorem

Document Type: Research Paper

Author

Oxford University Mathematical Institute

Abstract

Graham Higman published two important papers in 1960‎. ‎In the first of these‎ ‎papers he proved that for any positive integer $n$ the number of groups of‎ ‎order $p^{n}$ is bounded by a polynomial in $p$‎, ‎and he formulated his famous‎ ‎PORC conjecture about the form of the function $f(p^{n})$ giving the number of‎ ‎groups of order $p^{n}$‎. ‎In the second of these two papers he proved that the‎ ‎function giving the number of $p$-class two groups of order $p^{n}$ 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.

Keywords

Main Subjects


[1] B. Eick and E. A. O'Brien, Enumerating p -groups, J. Austral. Math. Soc. Ser. A. , 67 (1999) 191{205.
[2] B. Eick and M. Wesche, Enumeration of nilp otent asso ciative algebras of class 2 over arbitrary elds, J. Algebra ,
503 (2018) 573{589.
[3] J. A. Green, The characters of the nite general linear groups, Trans. Amer. Math. Soc. , 80 (1955) 402{447.
[4] G. Higman, Enumerating p -groups. I: Inequalities, Proc. London Math. Soc. , 10 (1960) 24{30.
[5] G. Higman, Enumerating p -groups. I I: Problems whose solution is PORC, Proc. London Math. Soc. , 10 (1960)
566{582.
[6] I. G. Macdonald, Symmetric functions and Hal l polynomials , The Clarendon Press, Oxford University Press, New
York, 1979.
[7] M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilp otent Lie rings whose order is the sixth
p ower of a prime, J. Algebra , 278 (2004) 383{401.
[8] M. Vaughan-Lee, On Graham Higman's famous PORC pap er, Int. J. Group Theory , 1 no. 4 (2012) 65{79.

[9] M. Vaughan-Lee, Enumerating algbras over a nite eld, In. J. Group Theory , 2 no. 3 (2013) 49{61.
[10] M. Vaughan-Lee, Choosing elements from nite elds , arXiv.1707.09652 (2017).
[11] B. Witty, Enumeration of groups of prime-power order , PhD thesis, Australian National University, 2006.