Hey there,

I'm having a bit of trouble with a practise question from my abstract algebra class.

The question is:

Let in $\displaystyle \sigma=$(1 2 3 4 5 6) in $\displaystyle S_6$. Show that $\displaystyle G=\{\epsilon,\sigma, \sigma^2,\sigma^3,\sigma^4,\sigma^5\}$ is a group using the operation of $\displaystyle S_6$. Is G abelian? How many elements $\displaystyle \tau$ of G satisfy $\displaystyle \tau^2=\epsilon$ and $\displaystyle \tau^3=\epsilon$?

I said that the operation of $\displaystyle S_6$ is closed and associative, that every element in $\displaystyle S_6$ has an inverse, and that $\displaystyle \epsilon=1_G$. I think this is enough to show that G is a group. However, I'm not sure what to do for the abelian or last part. I know that multiplication of disjoint cycles is abelian, but the elements in G that aren't the identity aren't disjoint. I'm kind of lost.

Any hints?