Fixed-point subgroup

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

$G^f=\{g\in G\mid f(g)=g\}.$

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 $f(g)=(g^T{)}^{-1}$ (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 s of S can be associated with the automorphism $g\mapsto sg{s}^{-1}$, i.e. conjugation by s. Then

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

that is, the centralizer of S.

See also

• Ring of invariants