automorphisminvariant

Automorphism-invariant is a term used in mathematics to describe a property that remains unchanged under automorphisms of a structure. Let X be a structure (such as a group, ring, graph, or topological space) with automorphism group Aut(X). A property P defined on elements, subobjects, or configurations of X is automorphism-invariant if, for every automorphism φ in Aut(X) and every object a on which P is defined, P(a) implies P(φ(a)). Equivalently, P is constant on the orbits of Aut(X) acting on the relevant objects.

In group theory, several classical properties are automorphism-invariant. For example, the order of an element is preserved by automorphisms, so the set of elements of a given order is automorphism-invariant. The center Z(G) and the derived subgroup [G,G] are invariant subgroups under automorphisms. More generally, invariants include any property that depends only on the isomorphism type or on orbit structure under Aut(G).

In graph theory, many vertex- or subgraph properties are automorphism-invariant. The degree of a vertex remains unchanged under automorphisms, and membership of a vertex in a particular Aut(G)-orbit is preserved. Properties such as connectivity or being part of a cycle are likewise preserved, since automorphisms preserve adjacency patterns.

Applications of automorphism-invariants include classifying structures up to isomorphism, understanding symmetry, and organizing objects into Aut(X)-orbits. They provide a language for describing features that cannot be altered by structural symmetry and play a fundamental role across algebra, combinatorics, and topology.