Prove the lemma from above.
Show that, for each m,n ÎN* and pÎSn , the permutations
p and pm are conjugate, if and
only if m
and each length of a cyclic factor of p are relatively prime.
Prove that the invertibility of the matrix
( gcd (i,k))i,k În
is equivalent to the following fact:
Two elements p, rÎSn are equivalent if and only if,
for each m ÎN*, the number of cyclic factors of pm and of rm are equal:
c( pm)=c( rm).
(Later on we shall return to this and give a proof of the regularity of
( gcd (i,k)). We shall in fact show that the determinant of this
matrix is f(1) ...f(n).)
Check the details in the equation.
Prove the Lemma.
last changed: August 28, 2001