Some congruences
According to Corollary we obtain from Theorem
the following results:
Corollary:
For any subgroups G <= Sn and H <= Sm, the following congruences
hold:
å pÎGmc( p) º0 ( | G | ),
åh ÎHa1(h)n º0 ( | H | ),
and also
å( r, p) ÎH ´G Õi=1na1( ri)ai( p) º0
( | H | | G | ),
as well as
å( y, p) ÎH wr G Õ n=1c( p)a1(hn( 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
xp=g is divisible by p.
In order to prove this we consider the action of Cp
on the set YX:=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, ...,gp) | g1 ...gp=g }.
As g forms its own conjugacy class, we obtain a subaction of Cp
on M (for example g1 ...gp is conjugate to gpg1 ...gp-1).
Hence the desired number k
of solutions of xp=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.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
