Some notation and definitions.

\(G\) is a finite group.

\(|G|\) is the order (size) of a group.

\(X\) is a finite set.

\(|X|\) is the order (size) of a set.

Group action :

For an action of the group \(G\) on the set \(X\), we have

i) \(g \land x \in X,\) for all \(g \in G, x \in X\).

An element of \(G \,(g \in G)\) acts on \((g \land x)\) an element of \(X \, (x \in X)\) which also belongs to \(X\).

ii) \(e \land x = x\), for all \(x \in X\), where \(e\) is the identity element of \(G\).

iii) \((gh) \land x = g \land (h \land x)\), for all \(g, h \in G\) and \(x \in X\).

\(gh\) is not commutative, \(h\) acts on \(x\) first.

Definition : Orbit

If a group \(G\) acts on a set \(X\) and \(x\) is an element of \(X\), then the orbit of \(x\) under \(G\) is the set of elements of \(X\) obtained by acting on \(x\) with the elements of \(G\). It is denoted by

Orb\((x) = \{g \land x : g \in G\}\)

The orbit of an element of \(X\) is a subset of \(X\).

Definition : Stabiliser

If a group \(G\) acts on a set \(X\) and \(x\) is an element of \(X\), then the stabiliser of \(x\) under \(G\) is the set of elements of \(G\) which fix \(x\). It is denoted by

Stab\((x) = \{g : g \in G,\, g \land x = x \}\)

The stabiliser of an element of \(X\) is a subgroup of \(G\).

The Orbit-Stabiliser Theorem :

When a finite group \(G\) acts on a finite set \(X\), then for each \(x \in X\)

|Orb\((x)|\) x |Stab\((x)| = |G|\)

When \(G = D_6\) and \(X = \{1, 2, 3, 4, 5, 6\}\)the group action table is

\(\begin{array}{c|cccccc|c}

\circ & 1 & 2 & 3 & 4 & 5 & 6 & \text{Cycle Permutation}\\

\hline

e&1&2&3&4&5&6&(1)(2)(3)(4)(5)(6)\\

r&2&3&4&5&6&1&(123456)\\

r^2&3&4&5&6&1&2&(135)(246)\\

r^3&4&5&6&1&2&3&(14)(25)(36)\\

r^4&5&6&1&2&3&4&(153)(264)\\

r^5&6&1&2&3&4&5&(165432)\\

s&1&6&5&4&3&2&(1)(4)(26)(35)\\

rs&2&1&6&5&4&3&(12)(36)(45)\\

r^2s&3&2&1&6&5&4&(2)(5)(13)(46)\\

r^3s&4&3&2&1&6&5&(14)(23)(56)\\

r^4s&5&4&3&2&1&6&(3)(6)(15)(24)\\

r^5s&6&5&4&3&2&1&(16)(25)(34)\\

\end{array}\)

© OldTrout \(2018\)

**No Audio file – it does not translate well**