Colourings of the n Gon |

Example:LetCdenote the following cyclic subgroup of_{p}sup:It acts on the setC_{p}:= á(1 ...p) ñ <= S_{ p}.X:=and hence, see the paragigmatic examples, also on the setp= {1, ...,p }Y, which can be considered as the set of all the colourings of the regular^{X}:=m^{ p}p-gon inmcolours. For example, in the case whenp:=5andm=2,Cacts on the set_{ 5}, consisting of all the 32 colourings of the regular pentagon with two colours (black and white), some of which are shown in the figure below.2^{ 5}Three colourings of the regular 5-gon.We now assume that

pis a prime. The Lemma shows thatCcontains, besides the identity element,_{ p}p-cycles only. The identity element ofCkeeps each_{ p}f Îfixed, while eachm^{ p}p-cycle fixes themmonochromatic colourings only (notice that(1 ...p)acts as a clockwise rotation of thep-gon after having numbered the vertices of thep-gon from 1 top, counterclockwise). Hence we obtain from the Cauchy-Frobenius Lemma thatprovided that| C_{p}\\m^{p}| =(1)/(p)(m^{p}+(p-1)m),pis a prime number. This implies thatm, and hence also^{p}+(p-1)mm, is congruent zero modulo^{p}-mp, for each positive integerm. It is clear that this is then also true for any integerz.In the case whenzis not divisible byp,we may even divide the differencezby^{p}-zz,obtainingFermat's CongruenceIfpis a prime number andzan integer which is prime top,thenz^{p-1}º1 (p).

This shows that group actions can also be useful in elementary number theory, and it seems appropriate to emphasize the following immediate implication of the Cauchy-Frobenius Lemma:

Later on we shall return to this result and we shall refine it considerably (numbers will be replaced by polynomials, and the congruence will be a congruence for the coefficients of the monomial summands). It is a very helpful tool for number theoretic purposes and shows again the efficiency of finite group actions.Corollary:For any action of a finite groupGon a finite setXwe have the congruenceå_{g ÎG}| X_{g}| º0 ( | G | ).

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Colourings of the n Gon |