Fixed-point subgroup

In algebra, the fixed-point subgroup formula_1 of an automorphism "f" of a group "G" is the subgroup of "G":
More generally, if "S" is a set of automorphisms of "G" (i.e., a subset of the automorphism group 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 "n"-by-"n" real matrices and formula_3 (called the Cartan involution). Then formula_1 is the group formula_5 of "n"-by-"n" orthogonal matrices.
To give an abstract example, let "S" be a subset of a group "G". Then each element "s" of "S" can be associated with the automorphism formula_6, i.e. conjugation by "s". Then
that is, the centralizer of "S".