# The Orbit-Stabiliser Theorem

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 $$2018$$

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