Fixed point subgroup

In algebra, the fixed-point subgroup G f 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 th 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 f ( g ) = ( g T ) − 1 (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 of a group G. Then each element of S can be thought of as an automorphism through conjugation. Then

that is, the centralizer of S.