Automorphism of a structure
Here, the word structure means a set with some additional data (or additional structure) provided to it. The automorphisms of the structure are those permutations of the set that preserve the additional structure. Loosely, these are the symmetries of the structure.
Some important facts:
- The composite of two automorphisms of a structure is also an automorphism of the structure. This is because, at both stages, the structure is preserved
- The inverse of an automorphism of a structure is also an automorphism of the structure
- The identity map is an automorphism of any structure
Thus, the automorphisms of any structure form a group -- a subgroup of the symmetric group on the set.
Automorphisms in a model theory
Automorphisms in a concrete category
Set with no additional structure
The automorphism group of a set with no additional structure is precisely the whole symmetric group.
Set with algebraic structure
- The automorphism group of a group is the collection of group automorphisms -- those bijections that preserve the group multiplication.
- The automorphism group of a vector space is precisely the general linear group over that vector space.
Set with combinatorial structure
We can talk of the automorphism group of a graph, of a set with some additional colouring or other structure etc.
Set with topological structure
We can talk of the self-homeomorphism group of a topological space. Here, the self-homeomorphisms play the role of automorphisms.
We can also talk of the self-diffeomorphism group of a differential manifold.