Automorphism group

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.

Examples

If X is a set with no additional structure, then any bijection from X to itself is an automorphism, and hence the automorphism group of X in this case is precisely the symmetric group of X. If the set X has additional structure, then it may be the case that not all bijections on the set preserve this structure, in which case the automorphism group will be a subgroup of the symmetric group on X. Some examples of this include the following:

• The automorphism group of a field extension $L/K$ is the group consisting of field automorphisms of L that fix K. If the field extension is Galois, the automorphism group is called the Galois group of the field extension.
• The automorphism group of the projective n-space over a field k is the projective linear group $\operatorname {PGL} _{n}(k).$
• The automorphism group $G$ of a finite cyclic group of order n is isomorphic to $(\mathbb {Z} /n\mathbb {Z} )^{\times }$, the multiplicative group of integers modulo n, with the isomorphism given by ${\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}$. In particular, $G$ is an abelian group.
• The automorphism group of a finite-dimensional real Lie algebra ${\mathfrak {g}}$ has the structure of a (real) Lie group (in fact, it is even a linear algebraic group: see below). If G is a Lie group with Lie algebra ${\mathfrak {g}}$, then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of ${\mathfrak {g}}$.

If G is a group acting on a set X, the action amounts to a group homomorphism from G to the automorphism group of X and conversely. Indeed, each left G-action on a set X determines $G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x$, and, conversely, each homomorphism $\varphi :G\to \operatorname {Aut} (X)$ defines an action by $g\cdot x=\varphi (g)x$. This extends to the case when the set X has more structure than just a set. For example, if X is a vector space, then a group action of G on X is a group representation of the group G, representing G as a group of linear transformations (automorphisms) of X; these representations are the main object of study in the field of representation theory.

Here are some other facts about automorphism groups:

• Let $A,B$ be two finite sets of the same cardinality and $\operatorname {Iso} (A,B)$ the set of all bijections $A\mathrel {\overset {\sim }{\to }} B$. Then $\operatorname {Aut} (B)$, which is a symmetric group (see above), acts on $\operatorname {Iso} (A,B)$ from the left freely and transitively; that is to say, $\operatorname {Iso} (A,B)$ is a torsor for $\operatorname {Aut} (B)$ (cf. #In category theory).
• Let P be a finitely generated projective module over a ring R. Then there is an embedding $\operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)$, unique up to inner automorphisms.

In category theory

Automorphism groups appear very naturally in category theory.

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.)

If $A,B$ are objects in some category, then the set $\operatorname {Iso} (A,B)$ of all $A\mathrel {\overset {\sim }{\to }} B$ is a left $\operatorname {Aut} (B)$-torsor. In practical terms, this says that a different choice of a base point of $\operatorname {Iso} (A,B)$ differs unambiguously by an element of $\operatorname {Aut} (B)$, or that each choice of a base point is precisely a choice of a trivialization of the torsor.

If $X_{1}$ and $X_{2}$ are objects in categories $C_{1}$ and $C_{2}$, and if $F:C_{1}\to C_{2}$ is a functor mapping $X_{1}$ to $X_{2}$, then $F$ induces a group homomorphism $\operatorname {Aut} (X_{1})\to \operatorname {Aut} (X_{2})$, as it maps invertible morphisms to invertible morphisms.

In particular, if G is a group viewed as a category with a single object * or, more generally, if G is a groupoid, then each functor $F:G\to C$, C a category, is called an action or a representation of G on the object $F(*)$, or the objects $F(\operatorname {Obj} (G))$. Those objects are then said to be $G$-objects (as they are acted by $G$); cf. $\mathbb {S}$-object. If $C$ is a module category like the category of finite-dimensional vector spaces, then $G$-objects are also called $G$-modules.

Automorphism group functor

Let $M$ be a finite-dimensional vector space over a field k that is equipped with some algebraic structure (that is, M is a finite-dimensional algebra over k). It can be, for example, an associative algebra or a Lie algebra.

Now, consider k-linear maps $M\to M$ that preserve the algebraic structure: they form a vector subspace $\operatorname {End} _{\text{alg}}(M)$ of $\operatorname {End} (M)$. The unit group of $\operatorname {End} _{\text{alg}}(M)$ is the automorphism group $\operatorname {Aut} (M)$. When a basis on M is chosen, $\operatorname {End} (M)$ is the space of square matrices and $\operatorname {End} _{\text{alg}}(M)$ is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, $\operatorname {Aut} (M)$ is a linear algebraic group over k.

Now base extensions applied to the above discussion determines a functor: namely, for each commutative ring R over k, consider the R-linear maps $M\otimes R\to M\otimes R$ preserving the algebraic structure: denote it by $\operatorname {End} _{\text{alg}}(M\otimes R)$. Then the unit group of the matrix ring $\operatorname {End} _{\text{alg}}(M\otimes R)$ over R is the automorphism group $\operatorname {Aut} (M\otimes R)$ and $R\mapsto \operatorname {Aut} (M\otimes R)$ is a group functor: a functor from the category of commutative rings over k to the category of groups. Even better, it is represented by a scheme (since the automorphism groups are defined by polynomials): this scheme is called the automorphism group scheme and is denoted by $\operatorname {Aut} (M)$.

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

See also

• Outer automorphism group
• Level structure, a technique to remove an automorphism group
• Holonomy group

References

• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. ISBN 9780691081014. MR 0349811. Zbl 0237.18005.
• Waterhouse, William C. (2012) [1979]. Introduction to Affine Group Schemes. Graduate Texts in Mathematics. Vol. 66. Springer Verlag. ISBN 9781461262176.

External links

• https://mathoverflow.net/questions/55042/automorphism-group-of-a-scheme