The automorphism group of an object 
, denoted 
, is the group of bijective
maps from 
to itself that preserve its structure and whose inverses preserve that structure
(Lang 2002). The automorphism group of a group preserves
the multiplication table, the automorphism
group of a graph the incidence
matrix, and that of a field the addition
and multiplication tables.
The graph automorphism group of a graph 
can be computed in the Wolfram Language
using GraphAutomorphismGroup[g].
A further realization theorem comes from the Yang-Baxter equation. Militaru (2026) proved that every group is the automorphism group of a set-theoretic solution, with a finite underlying set when the group is finite. The solutions need not satisfy a nondegeneracy restriction. Militaru (2026) reports AI assistance with the mathematical construction and proof.
