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 $f(g)=(g^{T})^{-1}$. Then $G^{f}$ is the group $O(n)$ of n-by-n orthogonal matrices.

The same definition applies to rings as well. Let R be a ring and f an automorphism of R. Then the subring fixed by f is the subring of R:

$R^{f}=\{r\in R\mid f(r)=r\}.$

Slightly more generally, if G is a subgroup of the automorphism group $\operatorname {Aut} (R)$ of R, then $R^{G}$, the intersection of $R^{g},\,g\in G$ is a subring called the subring fixed by H or, more commonly, the ring of invariants. A basic example appears in Galois theory; see Fundamental theorem of Galois theory.