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.

[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.

[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.

December 2019

Pages 11-28

**Receive Date:**17 August 2018**Revise Date:**16 October 2018**Accept Date:**26 September 2018**Published Online:**01 December 2019