Fixed-point subgroup

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

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

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.