Largest automorphism groups of simple algebras

Prove that over the complex numbers the maximal order of the automorphism group of a simple n-dimensional evolution algebra is $\max\{n!,n(2^n-1)\}$.

Background

The cyclic evolution algebra realizes the value n(2n−1)n(2^n-1), while a generic complete-graph construction realizes n!n!. The conjecture has been checked computationally through dimension four, but no general upper bound is known that simultaneously controls the diagonal and permutation parts of the automorphism group.

References

Over $\mathbb C$, the maximal order of the automorphism group of a simple $n$-dimensional evolution algebra is $\max{n!,\;n(2n-1)}$.

— Evolution algebras: a survey of results and a list of open problems  (2610.22248 - Kaygorodov et al., 15 Oct 2026) in Conjecture c:maxsimple, Section 2