Fixed-point subgroup

{{short description|Algebraic expression}}

{{onesource|date=September 2015}}

In algebra, the fixed-point subgroup $G^f$ of an automorphism f of a group G is the subgroup of G:{{Cite journal |last=Checco |first=James |last2=Darling |first2=Rachel |last3=Longfield |first3=Stephen |last4=Wisdom |first4=Katherine |date=2010 |title=On the Fixed Points of Abelian Group Automorphisms |url=https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1126&context=rhumj |journal=Rose-Hulman Undergraduate Mathematics Journal |volume=11 |issue=2 |pages=50}}

:$G^f\; =\; \backslash \{\; g\; \backslash in\; G\; \backslash mid\; f(g)\; =\; g\; \backslash \}.$

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 $g\; \backslash mapsto\; sgs^\{-1\}$, i.e. conjugation by s. Then

:$G^S\; =\; \backslash \{\; g\; \backslash in\; G\; \backslash mid\; sgs^\{-1\}\; =\; g\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\backslash \}$;

that is, the centralizer of S.