# Automorphism group

In mathematics, the **automorphism group** of an object *X* is the group consisting of automorphisms of *X*. For example, if *X* is a finite-dimensional vector space, then the automorphism group of *X* is the general linear group of *X*, the group of invertible linear transformations from *X* to itself.

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is called a **transformation group** (especially in old literature).

If *X* is an object in a category, then the automorphism group of *X* is the group consisting of all the invertible morphisms from *X* to itself. It is the unit group of the endomorphism monoid of *X*. (For some examples, see PROP.)

In general, however, an automorphism group functor may not be represented by a scheme.