Four dimensional homogeneous algebras (Q1073892)

An algebra is homogeneous if the automorphism group acts transitively on the one dimensional subspaces of the algebra. The purpose of this paper is to determine all homogeneous algebras of dimension 4. It continues previous work of the authors in which all homogeneous algebras of dimensions 2 and 3 were described. The main result is the proof that the field must be GF(2) and the algebras are of a type previously described by Kostrikin: letting \(\mu\) be any fixed element of \(GF(2^ n)\), define a new multiplication in \(GF(2^ n)\) by \(x\circ y=\mu (xy)^ 2\). There are 5 non-isomorphic algebras of dimension 4; a description of each is given and the automorphism group is calculated in each case.