Sets

Hi
I'd be grateful if someone could help me out with these questions on sets :smile:

.

Define the sign of the permutation σ by

sign σ = 1 if σ is even
sign σ = -1 if σ is odd

Verify that sign στ = sign σ sign τ for all τ,σ in Sn.

Which of the following is true?
a. Every permutation of odd order is an even permutation.
b. Every permutation of even order is an odd permutation.
Reasons why would be good too.

Thanks

Reply 1

Retropmot

for the first bit consider cases eg,
if σ is even and τ is odd what is στ
then just relate that to the signs

for the second bit consider how you could write the disjoint cycles (what has to be true about the length of these cycles) as transpostions and how many are there?

and for part b think counter example

although maybe you should find a better way of getting anwsers for problem sheet questions and maybe not leaving them till the last day would help

Reply 2

mikesgt2

This is professor Haydon's problem sheet isn't it. Retropmot makes a good point, if you can't do a question posting it on here and getting someone else to answer it for you is just cheating yourself.