Automorphism group

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An automorphism group Aut(X) is the set of all automorphisms of an object X, with the operation of composing maps. An automorphism is a structure‑preserving, invertible map from X to itself. For example, the automorphism group of a finite‑dimensional vector space is GL(V), the group of all invertible linear transformations. If X is a group, Aut(X) is the group of all group automorphisms. In geometry, automorphism groups are often called symmetry groups.

If X has no extra structure, every bijection X→X is an automorphism, so Aut(X) is the full symmetric group on X. When X has extra structure, only those bijections that preserve the structure are allowed, so Aut(X) is usually a subgroup of the symmetric group.

Group actions and representations
- If a group G acts on a set X, this is the same as giving a homomorphism G→Aut(X). Conversely, a homomorphism φ: G→Aut(X) defines an action by g·x = φ(g)(x).
- If X is a vector space, a G‑action is a representation of G as linear transformations of X.

A category theory perspective
- In a category, Aut(X) is the group of all invertible morphisms X→X. It sits as the units of the endomorphism monoid End(X).
- For objects A and B, Iso(A,B) (the set of isomorphisms A→B) behaves like a torsor for Aut(B): choosing a “base point” in Iso(A,B) is the same as choosing a particular way to identify B with the target of isomorphisms.

Functors and actions
- If F maps an object X1 in one category to X2 in another, F sends automorphisms of X1 to automorphisms of X2, giving a homomorphism Aut(X1)→Aut(X2).
- Viewing a group G as a one‑object category, a functor from G to a category C is the same as a G‑action (or G‑representation) on the image of the single object.

G‑objects and modules
- In a “module” category like vector spaces, a G‑object is the same as a G‑module: a space with a compatible G‑action.

Automorphism groups of algebraic structures
- Let M be a finite‑dimensional vector space over a field k with extra algebraic structure (such as an associative or Lie algebra structure). End_alg(M) is the set of structure‑preserving linear maps M→M, and Aut(M) is its group of invertible elements.
- When you pick a basis, End(M) looks like matrices, and End_alg(M) is defined by polynomial equations. Aut(M) becomes a linear algebraic group.
- If you allow base change to other rings R, the construction gives a group functor R ↦ Aut(M⊗R). This functor is often represented by a scheme, the automorphism group scheme Aut(M). In general, however, an automorphism group functor may not be representable by a scheme.

In short, automorphism groups capture the symmetries of mathematical objects, and they can be studied in many languages—set theory, group actions, category theory, and even algebraic geometry—depending on the structure you start with.