For finding, constructing and proving results about automorphisms and applying them into different contexts. An automorphism is an isomorphism from an object to itself, and collectively they form a group under composition of mappings. Sometimes the only automorphism is the trivial one (the identity map), but often structures will have interesting/non-trivial automorphisms.
Automorphisms in abstract algebra, topology, etc. are isomorphisms of a structure (monoid, group, ring, topological space, graph, etc.) with itself. More generally the concept can be defined on categories, where automorphism means an endomorphism (arrow from object to itself) which is an isomorphism (has an inverse arrow). The idea can be usefully applied in an astonishing variety of contexts.
Collectively the automorphisms of an object will form a group under composition (assuming for convenience such a collection is a set rather than a proper class). The smallest such a group can be is the trivial group, consisting solely of an object's identity map. Where non-trivial automorphisms exist, they are sometimes interpreted as the symmetries of that object, i.e. the self-mappings which conserve its essential properties.
