Enumerating algebras over a finite field

Document Type : Research Paper


Oxford University Mathematical Institute


‎We obtain the PORC formulae for the number of non-associative algebras‎ ‎of dimension 2‎, ‎3 and 4 over the finite field GF$(q)$‎. ‎We also give some‎ ‎asymptotic bounds for the number of algebras of dimension $n$ over GF$(q)$‎.


Main Subjects

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 (1979). Symmetric functions and {H}all polynomials. 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 H.~P. Petersson and M. Scherer (2004). The number of nonisomorphic two-dimensional algebras over a finite field. Results Math.. 45, 137-152 M. Vaughan-Lee (2012). Graham {H}igman's {PORC} conjecture. Jahres. Dtsch. Math.-Ver.. 114, 89-106 M. Vaughan-Lee (2012). On {G}raham {H}igman's famous {PORC} paper. Int. J. Group Theory. 1 (4), 65-79
Volume 2, Issue 3 - Serial Number 3
September 2013
Pages 49-61
  • Receive Date: 04 December 2012
  • Revise Date: 14 January 2013
  • Accept Date: 14 January 2013
  • Published Online: 01 September 2013