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In algebra, the **fixed-point subgroup** of an automorphism *f* of a group *G* is the subgroup of *G*:^{[1]}

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 (called the Cartan involution). Then is the group 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 , i.e. conjugation by *s*. Then

- ;

that is, the centralizer of *S*.

**^**Checco, James; Darling, Rachel; Longfield, Stephen; Wisdom, Katherine (2010). "On the Fixed Points of Abelian Group Automorphisms".*Rose-Hulman Undergraduate Mathematics Journal*.**11**(2): 50.