Fixed-point subgroup explained

In algebra, the fixed-point subgroup

G ^{f} of an automorphism f of a group G is the subgroup of G:^[1] G ^{f} = \{ g \in G \mid f(g) = 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 G^S.

For example, take G to be the group of invertible n-by-n real matrices and

f(g) = (g ^{T)} ^{-1} (called the Cartan involution). Then G ^{f} is the group O(n) 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 \mapsto sgs ^{-1} , i.e. conjugation by s. ThenG ^{S} = \{ g \in G \mid sgs ^{-1} = g forall s \in S\}

that is, the centralizer of S.

See also

• Ring of invariants

Notes and References

1. 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.