In algebra, the fixed-point subgroup of an automorphism f of a group G is the subgroup of G:^[1]

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 (called the Cartan involution). Then is the group 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 , i.e. conjugation by s. Then

;

that is, the centralizer of S.

• Ring of invariants