The Orbit-Stabiliser Theorem

OldTrout, Going Postal
In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to the tetrahedron to which g sends the chosen tetrahedron.

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