Fixed-point subgroup

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

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 GS.

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 of S can be thought of as an automorphism through conjugation. Then


that is, the centralizer of S.


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