Define a permutation $\sigma \in S_7$ by letting $\sigma (1) = 3, \sigma(2)=2, \sigma(3) = 7, \sigma(4)=5, \sigma(6)=1, \sigma (7)=6$.
I need help with presenting $\sigma$ as a product of disjoint cycles. And calculating the sign of $\sigma$.
I am really stuck on how to do this. Any help would be great.
The first cycle will be of the form $(1,\sigma(1) ,\sigma\sigma(1) ,\sigma\sigma\sigma(1),...)$. In other words, our first cycle will be (1376). Next, start with the lowest element not in the first cycle, which is $2$, and do the same process. Our second cycle will take the form $(2, \sigma(2) ,\sigma\sigma(2) ,\sigma\sigma\sigma(2),...)$. In other words, our second cycle will be $(2)$. Continuing this process, our third cycle will be $(4 ,\sigma(4), \sigma\sigma(4), \sigma\sigma\sigma(4),...)$. Our third cycle will be $(45)$. In conclusion $\sigma=(1376)(45)$.