Fixed-point subgroup

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

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

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

that is, the centralizer of S.