Group actions: orbits

Section 1.18 Group actions: orbits

Definition 1.18.1. Orbits.

Let \(G\times A\rightarrow A\) be a group action. Given \(a\in A\text{,}\) we define its orbit \(O_a\) as

\begin{equation} O_a=\{a'\mid a'=g\cdot a \text{ for some } g\in G\}=\{g\cdot a\mid g\in G\}\text{.}\tag{1.18.1} \end{equation}

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Definition 1.18.2. Transitive group action.

A group action \(G\times A\rightarrow A\) is transitive if for all \(a,a'\in A\) there is a \(g\in G\) satisfying \(a'=g\cdot a\text{.}\)

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Theorem 1.18.3. Orbit-stabilizer.

Let \(G\times A\rightarrow A\) be a group action.

For all \(a\in A\text{,}\) the function

\begin{align*} f\colon G/G_a \amp \rightarrow O_a\\ gG_a \amp \mapsto g\cdot a \end{align*}

is an isomorphism of \(G\)-sets, where \(G\) acts on \(G/G_a\) by left multiplication, and the action of \(G\) on \(O_a\) is given by restriction.

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For all \(a\in A\) we have

\begin{equation} \abs{O_a}=[G\colon G_a]\text{.}\tag{1.18.2} \end{equation}

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Proof.

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Example 1.18.4. Rotational symmetries of a tetrahedron.

Let \(T\) be a regular tetrahedron in \(\R^3\) and let \(G\) be the group of rotations in \(\R^3\) that map \(T\) to itself. Establish an isomorphism between \(G\) and one of our familiar groups.

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Solution.

Left as an exercise.

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Example 1.18.5. Rotational symmetries of a tetrahedron.

Let \(C\) be a cube in \(\R^3\) and let \(G\) be the group of rotational symmetries of \(C\text{.}\) Establish an isomorphism between \(G\) and one of our familiar groups.

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Solution.

Left as an exercise.

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Theorem 1.18.6. Orbit decomposition theorem.

Let \(G\times A\rightarrow A\) be a group action.

The relation \(a\sim a'\) if and only if \(a'\in O_a\) is an equivalence relation whose equivalence classes are the distinct orbits \(O_a\) of the group action. As a result, we have a partition

\begin{equation*} A=\bigcup_{a\in A}O_a \end{equation*}

of \(A\) into disjoint orbits, and \(O_a=O_{a'}\) if and only if \(a'\in O_a\text{.}\)

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As a \(G\)-set, \(A\) decomposes into a disjoint union of orbits, each of which is isomorphic as a \(G\)-set to \(G/H\) for some subgroup \(H\leq G\text{.}\)
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Proof.

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We end this section with a fantastic result commonly known as Burnside’s lemma even though it was first proved (using modern group-theoretic language) by Frobenius. Consequently, some mathematical wags refer to the result as the lemma that is not Burnside’s. We confuse things even further by giving it the designation of a theorem! The result allows us to count the distinct orbits of a group action in terms of so-called fix sets, which we now define.

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Definition 1.18.7. Fix sets.

Let \(G\times A\rightarrow A\) be a group action. Given an element \(g\in G\text{,}\) the fix set of \(g\) under this action, denoted \(\Fix(g)\text{,}\) is defined as

\begin{equation} \Fix(g)=\{a\in A\mid g\cdot a=a\}\text{.}\tag{1.18.3} \end{equation}

More generally, given a subset \(X\subseteq G\text{,}\) we define its fix set \(\Fix(X)\) as

\begin{equation} \Fix(X)=\{a\in A\mid g\cdot a=a \text{ for all } g\in X\}=\bigcap_{g\in X}\Fix(g)\text{.}\tag{1.18.4} \end{equation}

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Theorem 1.18.8 (Frobenius). Burnside’s lemma.

Let \(G\) be a finite group acting on a finite nonempty set \(A\text{,}\) and let \(O\) be the set of orbits of the group action. We have

\begin{equation} \abs{O}=\frac{1}{\abs{G}}\sum_{g\in G}\Fix(g)\text{.}\tag{1.18.5} \end{equation}

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Proof.

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