3.14 \(G\)-sets and the orbit-stabilizer theorem
In this section \(G\) is a group with identity element \(1\).
A \(G\)-set is an ordered pair \((X,\rho )\), where \(X\) is a set and \(\rho :G\to A(X)\) is an action of \(G\) on \(X\).
We have a few examples of \(G\)-sets already:
\(S_n\) acting on \([n]\).
\(\operatorname{\mathbf{GL}}_2(\mathbb {R})\) acting on \(\mathbb {R}^2\).
\(G\) acting on itself on the left.
\(G\) acting on itself by inner autorphisms.
One of the nice things about \(G\)-sets is that, once we have one \(G\)-set \(X\), we can often get many others. Here’s one example.
Suppose \(X\) is a \(G\)-set, and write \(\mathcal{P}(X)\) for the power set of \(X\). I.e., \(\mathcal{P}(X)\) is the set of all subsets of \(X\). For each \(g\in G\) and \(S\subset X\), set \(gS = \{ gs: s\in S\} \). Then the map \(a:G\times \mathcal{P}(X)\to \mathcal{P}(X)\) given by \(a(g,S) = gS\) defines an action of \(G\) on \(\mathcal{P}(X)\).
Exercise (maybe obvious).
If \(X\) is a \(G\)-set and \(Y\subseteq X\), then we say that \(Y\) is a sub \(G\)-set if \(gy\in Y\) for all \(y\in Y\).
Clearly, a sub \(G\)-set of a \(G\)-set \(X\) is itself a \(G\)-set.
Suppose \(H\leq G\). The \(G/H\) is a sub \(G\)-set of \(\mathcal{P}(G)\).
Suppose \(g_1,g_2\in G\). Then \(g_1(g_2H) = (g_1g_2)H\in G/H\).
The \(G\)-set \(G/H\) is so important that, unless we
We say that a \(G\)-set \(X\) is transitive if \(G\backslash X\) is a singleton. In other words, \(X\) is transitive if it has exactly one orbit.