# Automorphism group

## Mathematical group formed from the automorphisms of an object / From Wikipedia, the free encyclopedia

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In mathematics, the **automorphism group** of an object *X* is the group consisting of automorphisms of *X* under composition of morphisms. For example, if *X* is a finite-dimensional vector space, then the automorphism group of *X* is the group of invertible linear transformations from *X* to itself (the general linear group of *X*). If instead *X* is a group, then its automorphism group $\operatorname {Aut} (X)$ is the group consisting of all group automorphisms of *X*.

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a **transformation group**.

Automorphism groups are studied in a general way in the field of category theory.