Fixed-point subgroup

Short description: Algebraic expression

In algebra, the fixed-point subgroup [math]\displaystyle{ G^f }[/math] of an automorphism f of a group G is the subgroup of G:

[math]\displaystyle{ G^f = \{ g \in G \mid f(g) = g \}. }[/math]

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 [math]\displaystyle{ f(g)=(g^T)^{-1} }[/math] (called the Cartan involution). Then [math]\displaystyle{ G^f }[/math] is the group [math]\displaystyle{ O(n) }[/math] 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 [math]\displaystyle{ g \mapsto sgs^{-1} }[/math], i.e. conjugation by s. Then

[math]\displaystyle{ G^S = \{ g \in G \mid sgs^{-1} = g \text{ for all } s \in S\} }[/math];

that is, the centralizer of S.