Abstract Algebra - Action, Orbit and Stabilizers

Action

An action of a group \(G\) on a set \(X\) is a homomorphism from \(G\) to \(S_X\).

💡 We imagine the elements of \(G\) permuting the points of \(X\) in a way that is compatible with the algebraic structure of \(G\).

Orbit

Given an action of \(G\) on \(X\) and a point \(x \in X\), the set of all images \(g(x)\), as \(g\) varies through \(G\), is called the orbit of \(x\) and written \(G(x)\).

Stabilizer

If \(x\) is a point of \(X\), the elements of \(G\) which leave \(x\) fixed form a subgroup of \(G\) called the stabilizer \(G_x\) of \(x\).

Some examples

Example 1

The infinite cyclic group \(\mathbb{Z}\) acts on the real line by translation. The integer \(n \in \mathbb{Z}\) sends the real number \(x\) to \(n + x\). If \(m\) and \(n\) are integers, then

\[ (m + n) + x = m + (n + x) \]

This is an action. The orbit of \(x\) consists of all translates \(n + x\) where \(n \in \mathbb{Z}\). The stabilizer is \(\{0\}\) since \(0+x = x\) for all \(x \in \mathbb{R}\).

Example 2

Let \(n \in \mathbb{Z}\) sends \(x \in \mathbb{R}\) to \((-1)^nx\). The permutation associated to every even integer is the identity permutation of \(\mathbb{R}\), and that associated to all the odd integers is \(x \to -x\). Since \((-1)^{m + n} = (-1)^m(-1)^n\) for every two integers \(m\) and \(n\). The orbit of \(x\) is just \(\{-x, x \}\) where \(x \neq 0\), otherwise is \(\{0\}\). The stabilizer is \(2\mathbb{Z}\) if \(x \neq 0\) and \(\mathbb{Z}\) when \(x = 0\).

So this is also an action.

💡 We can see from the above examples that a same group \((\mathbb{Z}, +)\) can action on a set \(X\) in different ways.

Theorems

Theorem 1

Points in the same orbit have conjugate stabilizers.

💡 Suppose \(x\) and \(y\) belong to the same orbit, say \(g(x) = y\), then we have \(gG_xg^{-1} = G_y\).

Orbit-Stabilizer Theorem

For each \(x \in X\), the correspondence \(g(x) \to gG_x\) is a bijection between \(G(x)\) and the set of let cosets of \(G_x\) in \(G\).

\[ G(x) \cong G/G_x \]

💡 Consider the similarity between the Orbit-Stabilizer Theorem and the Fundemental Homomorphism Theorem.

Theorem 2

If \(G\) is finite, the size of each orbit is a divisor of the order of \(G\).

\[ |G(x)| = |G|/|G_x| \]

The Counting Theorem

Suppose we have an action of a finite group \(G\) on a set \(X\). Write \(X^g\) for the subset of \(X\) consisting of those points which are left fixed by the element \(g\) of \(G\). The number of distinct orbits, denoted by \(|X/G|\), is