Group actions

Let S be a set. A bijection $f:S\longrightarrow S$ is called a permutation. Let $Sym(S)=\{$permutations on S$\}.$ If $f\in Sym(S)$ and $\alpha \in S,$ then $f(\alpha )$ denotes the image of $\alpha$ under f. The set Sym(S) forms a group under composition of mappings, called the symmetric group on S.

We say that the group G acts on the set S if there is a homomorphism $T:G\longrightarrow Sym(S).$ For $\alpha$ in S and g in G we abbreviate by writing $g(\alpha )$ in place of $T(g)(\alpha );$ if h is also in G we have $(hg)(\alpha )=T(hg)(\alpha )=T(h)(T(g)(\alpha ))=h(g(\alpha ))$. If kerT=1 we say that G acts faithfully.

Define a subgroup $G_{\alpha}=\{g\in G:g(\alpha )=\alpha \}$ and call it the stabilizer of $\alpha$ in G. Further define $G(\alpha )=\{g(\alpha ):g\in G\}$ and call this subset of S the orbit of $\alpha$ under the action of G.

* Under a group action the set S decomposes into a disjoint union of orbits.

Indeed, pick an element $\alpha_1$ of S and produce its orbit under the action of G; pick another element $\alpha_2$ of S outside this orbit and produce its orbit (the two orbits are easily seen to be disjoint, since $h(\alpha_2 )=g(\alpha_1)$ implies $\alpha_2=h^{-1}g(\alpha ),$ a contradiction); proceed until S is exhausted.

* The cardinality (or length) of the orbit is equal to the index of the stabilizer. [Specifically, we assert that $\vert G(\alpha )\vert =\vert G:G_{\alpha}\vert .$]

This follows by observing that $\alpha$ is sent into $g(\alpha )$ by exacly those group elements that are in the coset $gG_{\alpha}$.

* (The class equation) $\vert S\vert =\sum_i\vert G :G_{\alpha_i}\vert ,$ where $\alpha_i$ is a representative from orbit i.

Indeed, S is the disjoint union of orbits. Its cardinality is, therefore, the sum of the lengths of these orbits, which are indices of stabilizers of orbit representatives.

* The stabilizers of two elements from the same orbit are conjugate in G. [Specifically, $G_{h(\alpha )}=hG_{\alpha}h^{ -1}.$]

The map $g\longrightarrow hgh^{-1}$ is a bijection between $G_{\alpha}$ and $G_{h(\alpha )}$, which allows us to conclude that $G_{h(\alpha )}=hG_{\alpha}h^{ -1}.$

The homomorphic image T(G) is called a permutation representation of the group G on the set S. We call T(G) a permutation group on S.

A permutation group on a set S is called transitive if for any two elements of S there exists an element of the group sending one into the other. When the group element g is viewed as a permutation, the elements of S that it fixes are called the fixed points of g. A transitive group is regular if the only element of the group which has fixed points is the identity.