Permutation Representations and Automorphisms of Evolution Algebras

Abstract

We prove that the natural permutation representation of highly transitive finite groups cannot be realized as the full automorphism group of an idempotent, finite-dimensional evolution algebra acting on the set of lines spanned by its natural elements. Specifically, for any suf- ficiently large integer n and k ≥ 4, there does not exist an idempotent evolution algebra X of dimension n such that Aut(X) is isomorphic to a proper k-transitive subgroup of Sn. Nevertheless, we show that for any finite group G, any permutation representation ξ : G → Sn, and any field k, there exists an idempotent, finite-dimensional evolution k-algebra X such that Aut(X) ∼= G, and the induced representation of Aut(X) on the natural idempotents of X is equivalent to ξ.