# Math Help - Disjoint cycles...

1. ## Disjoint cycles...

Q1) Write in the following as a single cycle or a product of disjoint cycles?

A) ( 1 2 3 ) ^(-1) ( 2 4 ) ( 1 2 3 )

B) ( 1 3 )^(-1) ( 2 4 ) ( 2 3 5 )^(-1)

I am having some difficulties understanding the difference between the normal set and this inverse sets...I cant figure it out..
I do this first one. First $(123)=(321)=(132)$ by the other post. Thus, we need to simplify $(132)(24)(123)$ remember this means $\sigma = (132)\circ (24) \circ (123)$. To do these problems we map $1$ and then find $\sigma(1)$ then $\sigma(\sigma(1))$ until we complete the cycle. Note, $1\to 2$ by applying $(123)$ first then $2\to 4$ by applying $(24)$ second and $4\to 4$ by applying $(132)$ third. Thus, $1\to 4$ by $\sigma$. Now we find $\sigma(4)$. Note $4\to 4$ by $(123)$ first then $4\to 2$ under $(24)$ and finally $2\to 1$ by $(132)$. Thus, $\sigma(\sigma(1)) = 1$ and it means we only write $(14)$ as we express $\sigma$ as a product of disjoint cycles. Next we do the same with $2$. See that $2\to 2$ under $\sigma$ and so $3\to 3$. Which means the representation is $(14)(2)(3) = (14)$.