In algebra, the fixed-point subgroup G f {\displaystyle G^{f}} of an automorphism f of a group G is the subgroup of G:

G f = { g ∈ G ∣ f ( g ) = g } . {\displaystyle G^{f}=\{g\in G\mid f(g)=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 GS.

For example, take G to be the group of invertible n-by-n real matrices and f ( g ) = ( g T ) − 1 {\displaystyle f(g)=(g^{T})^{-1}} (called the Cartan involution). Then G f {\displaystyle G^{f}} is the group O ( n ) {\displaystyle 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 s of S can be associated with the automorphism g ↦ s g s − 1 {\displaystyle g\mapsto sgs^{-1}} , i.e. conjugation by s. Then

G S = { g ∈ G ∣ s g s − 1 = g  for all  s ∈ S } {\displaystyle G^{S}=\{g\in G\mid sgs^{-1}=g{\text{ for all }}s\in S\}} ;

that is, the centralizer of S.