Automorphism of a first-order structure

An automorphism is a bijection from a structure to itself that preserves every operation and relation in its language.

The automorphism group $\mathrm{Aut}(M)$ consists of all automorphisms of $M$, with composition as its group operation.

After adjoining a unary function symbol $f$, injectivity, surjectivity, preservation of each operation, and preservation in both directions of each relation form an axiom scheme whose models interpret $f$ as an automorphism.

A family $(f_g{)}_{g\in G}$ of automorphisms satisfying $f_g{f}_h=f_{gh}$ defines a group action. It is faithful exactly when distinct indices define distinct automorphisms, equivalently when $G$ embeds in the automorphism group.

A first-order structure is rigid when its identity map is its only automorphism. Every finite linear order is rigid, whereas $(\mathbb{Z},<)$ is nonrigid because every translation is an order automorphism.

If a theory has arbitrarily large finite nonrigid models, expand its language by a unary function $f$, axiomatize that $f$ is an automorphism, require $f$ to move some element, and add sentences demanding at least $n$ elements for every $n$. Every finite fragment has a sufficiently large finite nonrigid model, so compactness supplies an infinite nonrigid model.