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Concept
Automorphism group
Summary
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 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 .
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 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
The automorphism group of a finite cyclic group of order n is isomorphic to , the multiplicative group of integers modulo n, with the isomorphism given by . In particular, is an abelian group.
The automorphism group of a finite-dimensional real Lie algebra 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 , then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of .
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.
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