Enumerating algebras over a finite field

Graham Higman wrote two immensely important and in uential papers on enumerating p-groups in the late 1950s. The papers were entitled Enumerating p-groups I and II, and were published in the Proceedings of the London Mathematical Society in 1960. A complete description of the algebras of dimension 2 over a nite eld is given by Petersson and Scherer. They also give formulae for the number of algebras of dimension 2 over GF(q), which agree with the formulae given here. We give the formulae for the numbers of algebras of dimensions 2, 3 and 4 in Section 2. In Section 3 we give a broad outline of how the formulae can be obtained and in Section 4 we draw some quite precise conclusions about the asymptotic form of the formulae for general n.