algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the fixed-point subgroup

G^f

of an

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

''f'' of a group ''G'' is the

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of ''G'': :

G^f = \.

More generally, if ''S'' is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of automorphisms of ''G'' (i.e., a subset of the

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 g ...

of ''G''), then the set of the elements of ''G'' that are left fixed by every automorphism in ''S'' is a subgroup of ''G'', denoted by ''G''^''S''. For example, take ''G'' to be the group of

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...

''n''-by-''n'' real matrices and

f(g)=(g^T)^

(called the Cartan involution). Then

G^f

is the group

O(n)

of ''n''-by-''n'' orthogonal matrices. To give an abstract example, let ''S'' be a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of a group ''G''. Then each element ''s'' of ''S'' can be associated with the automorphism

g \mapsto sgs^

, i.e.

conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

by ''s''. Then :

G^S = \

; that is, the centralizer of ''S''.

References

Algebraic groups {{group-theory-stub

See also

References