Self-diffeomorphism group

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological, analytic or discrete structure
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Definition

The self-diffeomorphism group of a differential manifold is defined in any of the following equivalent ways:

• It is the group whose elements are diffeomorphisms from the differential manifold to itself, and where multiplication is by composition
• It is the automorphism group of the differential manifold, viewed as an object in the category of differential manifolds with smooth maps

Facts

• The self-diffeomorphism group of a differential manifold acts transitively on the manifold.
• The self-diffeomorphism group is a subgroup of the self-homeomorphism group. It is often a conjugate-dense subgroup and often is also a finite-dominating subgroup.