Section 18.1 Pre-read
Suppose \(G\) is a group acting on a set \(X\text{.}\) Recall that the Stabilizer of set element \(x\) is the subgroup of those group elements \(g\) so that \(\grpact{g}{x} = x\text{.}\) There is a related dual concept to the concept of stabilizer.
Definition 18.1.1. Fixed set of a group element \(g\).
Write \(X^g\) or \(\Fix_X(g)\) to mean the collection of those set elements \(x\) so that \(\grpact{g}{x} = x\text{.}\)
Warning 18.1.2. Keep it straight!
A stabilizer is a subgroup of group elements from \(G\) that fix a single set element \(x\) in common, while a fixed set is a collection of set elements from \(X\) that are fixed in common by a single group element \(g\text{.}\)