Structure of quasi-invariant sets (Q811567)

Let \(G\) be a group acting on a set \(X\), so that \(x\mapsto x^ g\) for \(x\in X\), \(g\in G\). A subset \(A\subseteq X\) is said to be quasi-invariant if \(| A^ g\backslash A|\leq 1\) for every \(g\in G\). It is proved that a quasi-invariant subset is either an invariant subset or an invariant subset ``\(\pm\) a point''. More precisely, one of the following cases holds: (i) \(A\) is an invariant subset, i.e., \(A^ g=A\) for all \(g\in G\); (ii) \(\exists a\in A\) such that \(A\backslash \{a\}\) is an invariant subset; (iii) \(\exists x\in X\backslash A\) such that \(A\cup \{x\}\) is an invariant subset. The obtained structure is useful in some group theory problems.