automorphism group

English

Noun

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^[1], 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 $\aleph _{0}$-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.

Usage notes

• Especially in older literature, a subgroup of an automorphism group may be called a transformation group.
• The automorphism group of a field extension $L/K$ may be denoted ${\mbox{Aut}}(L/K)$; it consists of all automorphisms of L which leave K fixed.

Synonyms

• symmetric group
  • symmetry group (chiefly in geometry)

Hyponyms

• Galois group
• general linear group
• projective linear group

Translations

group whose elements are the automorphisms of a given mathematical object

See also

• group automorphism (type of automorphism)

Further reading

• Automorphism on Wikipedia.Wikipedia
• Automorphism group of a free group on Wikipedia.Wikipedia
• Inner automorphism 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