| | | The Cauchy-Frobenius Lemma 2 |
The Cauchy-Frobenius Lemma 2
Corollary:
Let GX be a finite action and let C denote a
transversal of the conjugacy classes of G. Then
| G \\X | =(1)/( | G | ) åg Î C
| CG(g) | | Xg | =
åg Î C | CG(g) | -1 | Xg | .
Here is the faster version of the
Cauchy-Frobenius Lemma.
Another formulation of the Cauchy-Frobenius Lemma makes use of the
permutation representation g -> bar (g) defined by the action
in question. (Actually in all our programs we apply this version of the Lemma.)
The permutation group bar (G) which is the image of G
under this representation, yields the action bar (G)X of
bar (G) on X, which has the same orbits, and so we also have:
Corollary:
If X denotes a finite G-set, then (for
any group G) the following identity holds:
| G \\X | =(1)/( | bar (G) | ) å bar (g)
Îbar (G) | X bar (g) | =(1)/( | bar (G) | )
å bar (g) Îbar ( C) | C bar (G)( bar (g))
| | X bar (g) | ,
where
bar ( C) denotes a transversal of the conjugacy classes of bar (G).
last changed: January 19, 2005
| | | The Cauchy-Frobenius Lemma 2 |