Some congruences

Some congruences

According to Corollary we obtain from Theorem the following results:

Corollary: For any subgroups G <= S_n and H <= S_m, the following congruences hold:

å _pÎGm^{c( p)} º0 ( | G | ), å_{h ÎH}a₁(h)ⁿ º0 ( | H | ),

and also

å_{( r, p) ÎH ´G} Õ_i=1ⁿa₁( rⁱ)^{a_i( p)} º0 ( | H | | G | ),

as well as

å_{( y, p) ÎH wr G} Õ _n=1^{c( p)}a₁(h_n( y, p)) º0 ( | H | ⁿ | G | ).

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 equation xⁿ=1 in a finite group G is divisible by n, if n divides the order of G. There are many proofs of this result and also many generalizations. Later on we return to this problem, at present we can only discuss a particular case which can be treated with the tools we already have at hand.

Example: Let g denote an element of a finite group which forms its own conjugacy class and consider a prime number p, which divides | G | . We want to show that the number of solutions x ÎG of the equation x^p=g is divisible by p. In order to prove this we consider the action of C_p on the set Y^X:=G ^p. The orbits are of length 1 or 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 subset M ÍG ^p:

M:= {(g₁, ...,g_p) | g₁ ...g_p=g }.

As g forms its own conjugacy class, we obtain a subaction of C_p on M (for example g₁ ...g_p is conjugate to g_pg₁ ...g_p-1). Hence the desired number k of solutions of x^p=g is equal to the number of orbits of length 1 in M. Now we consider the number l of orbits of length p in M. It satisfies the equation k+pl= | M | . As each equation g₁g'=g has a unique solution g₁ in G, we moreover have that | M | = | G | ^p-1. Thus

k ºk+pl= | M | º0 (p),

which completes the proof. We note in passing that the center of G consists of the elements which form their own conjugacy class, so that we have proved the following:

Corollary: If the prime p divides the order of the group G, then the number of p-th roots of each element in the center of G is divisible by p. In particular the number of p-th roots of the unit element 1 of G has this property (and it is nonzero, since 1 is a p-th root of 1), and hence G contains elements of order p.

This result can be used in order to give an inductive proof of Sylow's Theorem which we proved in example.

Some congruences