Some congruences |

Further congruences show up in the enumeration of group elements with prescribed properties. This theory of enumeration in finite groups is, besides the enumeration of chemical graphs, one of the main sources for the theory of enumeration which we are discussing here. A prominent example taken from this complex of problems is the following one due to Frobenius: The number of solutions of the equationCorollary:For any subgroupsG <= Sand_{n}H <= S, the following congruences hold:_{m}and alsoå_{ pÎG}m^{c( p)}º0 ( | G | ), å_{h ÎH}a_{1}(h)^{n}º0 ( | H | ),as well aså_{( r, p) ÎH ´G}Õ_{i=1}^{n}a_{1}( r^{i})^{ai( p)}º0 ( | H | | G | ),å_{( y, p) ÎH wr G}Õ_{ n=1}^{c( p)}a_{1}(h_{n}( y, p)) º0 ( | H |^{n}| G | ).

Example:Letgdenote an element of a finite group which forms its own conjugacy class and consider a prime numberp, which divides| G |. We want to show that the number of solutionsx ÎGof the equationxis divisible by^{p}=gp. In order to prove this we consider the action ofCon the set_{p}Y. The orbits are of length 1 or^{X}:=G^{ p}p. An orbit is of length 1 if and only if it consists of a single and therefore of a constant mapping(g', ...,g'), say. We now restrict our attention to the following subsetM ÍG:^{ p}AsM:= {(g_{1}, ...,g_{p}) | g_{1}...g_{p}=g }.gforms its own conjugacy class, we obtain a subaction ofCon_{p}M(for examplegis conjugate to_{1}...g_{p}g). Hence the desired number_{p}g_{1}...g_{p-1}kof solutions ofxis equal to the number of orbits of length 1 in^{p}=gM. Now we consider the numberlof orbits of lengthpinM. It satisfies the equationk+pl= | M |. As each equationghas a unique solution_{1}g'=ggin_{1}G, we moreover have that| M | = | G |. Thus^{p-1}which completes the proof. We note in passing that thek ºk+pl= | M | º0 (p),centerofGconsists of the elements which form their own conjugacy class, so that we have proved the following:This result can be used in order to give an inductive proof of Sylow's Theorem which we proved in example.Corollary:If the primepdivides the order of the groupG, then the number ofp-th roots of each element in the center ofGis divisible byp. In particular the number ofp-th roots of the unit element1ofGhas this property (and it is nonzero, since1is ap-th root of1), and henceGcontains elements of orderp.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Some congruences |