a) Let = (123)(456). Find an odd permutation 2 S6 such that =.
b) Let ` be an odd integer such that ` 3. Suppose that 2 Sn, jjis odd, and
in the disjoint cycle decomposition of , there are at least twodierent cycles of
length `. Prove that there exists an odd permutation 2 Sn suchthat = .
c) Note: This part will not be marked. Do not hand in a solutionto this part.
Suppose that jj is odd, (i) = i for at most one i, and all ofthe cycles in the
disjoint cycle decomposition of have distinct lengths. Provethat there is no
odd permutation
Let alpha = (123) (456). Find an odd permutation beta S6 such that alphabeta = betaalpha. Let l be an odd integer such that l 3. Suppose that alpha Sn, |alpha| is odd, and in the disjoint cycle decomposition of alpha, there are at least two different cycles of length l. Prove that there exists an odd permutation beta Sn such that alphabeta = betaalpha. Note: This part will not be marked. Do not hand in a solution to this part. Suppose that |alpha| is odd, alpha(i) = i for at most one i, and all of the cycles in the disjoint cycle decomposition of alpha have distinct lengths. Prove that there is no odd permutation in Sn that commutes with alpha. (That is, the centralizer of alpha is a subgroup of An.)
