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In algebra, the fixed-point subgroup G f {\displaystyle G^{f}} of an automorphism f of a group G is the subgroup of G:
More generally, if S is a set of automorphisms 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.
For example, take G to be the group of invertible n-by-n real matrices and f = − 1 {\displaystyle f=^{-1}} . Then G f {\displaystyle G^{f}} is the group O {\displaystyle O} 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