Editing Fixed point (mathematics) (section)

== Fixed point of a group action ==

In [[algebra]], for a group ''G'' acting on a set ''X'' with a [[group action]] <math>\cdot</math>, ''x'' in ''X'' is said to be a fixed point of ''g'' if <math>g \cdot x = x</math>.

The [[fixed-point subgroup]] <math>G^f</math> of an [[automorphism]] ''f'' of a [[group (mathematics)|group]] ''G'' is the [[subgroup]] of ''G'':
<math display="block">G^f = \{ g \in G \mid f(g) = g \}.</math>

Similarly, the [[fixed-point subring]] <math>R^f</math> of an [[automorphism]] ''f'' of a [[ring (mathematics)|ring]] ''R'' is the [[subring]] of the fixed points of ''f'', that is,
<math display="block">R^f = \{ r \in R \mid f(r) = r \}.</math>

In [[Galois theory]], the set of the fixed points of a set of [[field automorphism]]s is a [[field (mathematics)|field]] called the [[fixed field]] of the set of automorphisms.