automorphism group

automorphism group (plural automorphism groups)

1. (group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition.

   The automorphism group of a set is the set's symmetric group.

   The automorphism group of a finite-dimensional vector space (respectively, projective space) is the space's general linear group (respectively, projective linear group).

   • 1984, José M. Isidro, László L. Stachó, Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, Elsevier (North-Holland), page vi,

     Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds.

   • 1993, Matatyahu Rubin, The Reconstruction of Trees from Their Automorphism Groups, American Mathematical Society, page 5,

     Recently Droste, Holland and Macpherson in [DHM1], [DHM2] and [DHM3] investigated various questions concerning the automorphism groups of homogeneous ${\displaystyle \aleph _{0}}$ -categorical trees.

   • 2002, Masamichi Takesaki, Theory of Operator Algebras II, Springer, page 91,

     Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups.

• Especially in older literature, a subgroup of an automorphism group may be called a transformation group.
• The automorphism group of a field extension ${\displaystyle L/K}$ may be called its Galois group; it comprises automorphisms that preserve the elements of the base field, ${\displaystyle K}$ .
• Authors sometimes refer to the automorphism group of a given object type. This can be rendered more precisely as the group of automorphisms of an object of the given type.
  • In consequence of this usage, it might appear as though automorphism groups of different types of object are intrinsically different. In fact, most of the difference derives from differences in what comprises an automorphism for the various object types (more specifically, from what structures an automorphism is required to preserve).

• (group of automorphisms of a mathematical object): symmetric group
  • symmetry group (chiefly in geometry)

• (group of automorphisms of a mathematical object): general linear group, projective linear group

• group automorphism (type of automorphism)

• Automorphism on Wikipedia.Wikipedia
• Automorphism group of a free group on Wikipedia.Wikipedia
• Inner automorphism#Inner and outer automorphism groups on Wikipedia.Wikipedia
• Outer automorphism group on Wikipedia.Wikipedia
• Holonomy on Wikipedia.Wikipedia
• Level structure (algebraic geometry) on Wikipedia.Wikipedia
• Algebraic system, automorphism of an on Encyclopedia of Mathematics
• Automorphism Group on Wolfram MathWorld