The Number of Nonisomorphic Two-dimensional Algebras over a Finite Field

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In this paper, we derive explicit formulas for the number of nonisomorphic two-dimensional nonassociative algebras, possibly without a unit, over a finite field. The proof combines the first author’s general classification theory of two-dimensional nonassociative algebras over arbitrary base fields with elementary counting arguments which are primarily addressed to the problem of determining the number of orbits of a finite set acted upon by the group of integers mod 2. The number of nonisomorphic two-dimensional division algebras will also be determined.