Exercises

Exercises

E: Prove the lemma from above.

E: Show that, for each m,n ÎN^* and pÎS_n , the permutations p and p^m are conjugate, if and only if m and each length of a cyclic factor of p are relatively prime.

E: Prove that the invertibility of the matrix

( gcd (i,k))_{i,k În}

is equivalent to the following fact: Two elements p, rÎS_n are equivalent if and only if, for each m ÎN^*, the number of cyclic factors of p^m and of r^m are equal:

c( p^m)=c( r^m).

(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).)

E: Check the details in the equation.

E: Prove the Lemma.

E: Check corollary.

Exercises