E: Prove the lemma from above.
E: 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.
E: Prove that the invertibility of the matrix( gcd (i,k))i,k Înis 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).)
E: Check the details in the equation.
E: Prove the Lemma.
E: Check corollary.
|harald.fripertinger "at" uni-graz.at||http://www-ang.kfunigraz.ac.at/~fripert/||UNI-Graz||Institut für Mathematik||UNI-Bayreuth||Lehrstuhl II für Mathematik|