Exercises |

E:Prove the lemma from above.

E:Show that, for eachm,n ÎandN^{*}pÎS, the permutations_{n}pandpare conjugate, if and only if^{m}mand each length of a cyclic factor ofpare relatively prime.

E:Prove that the invertibility of the matrixis equivalent to the following fact: Two elements( gcd (i,k))_{i,k În}p, rÎSare equivalent if and only if, for each_{n}m Îthe number of cyclic factors ofN^{*},pand of^{m}rare equal:^{m}(Later on we shall return to this and give a proof of the regularity ofc( p^{m})=c( r^{m}).( gcd (i,k)).We shall in fact show that the determinant of this matrix isf(1) ...f(n).)

E:Check the details in the equation.

E:Prove the Lemma.

E:Check corollary.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Exercises |