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 | ) å_{g Î C} | C^G(g) | | X_g | = å_{g Î C} | C_G(g) | ^-1 | X_g | .

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).

The Cauchy-Frobenius Lemma 2