5.4: If sigma in S_n has cycle type n_1,...,n_r, what is sgn(sig)? (sgn is the sign homomorphism)
sgn(sigma) = 1 if sigma is even. sgn(sigma) = -1 is sigma is odd
cycle type is the length of the cycle type. If n_2 = 2, sigma has two 2-cycles.
The Attempt at a Solution
I know what cycle type is (if n_i = j, there are j cycles of length i), and sgn(sigma) is easy. How would I go about expanding (sig) to find how many two cycles I have, if that's even what I should be doing. It doesn't seem that worthwhile to generalize this for cycle type.
I was thinking of writing sgn(sig) as a product of sgn(sig_i) where (sig_i) is an individual cycle of the product of cycles forming (sig), but I don't think that exactly accounts for multiples.
I'm also not sure how to come up with conditions that will say, depending on r, if sgn(sig) = 1, or sgn(sig) = -1.