Please could a member run me through this question and help me understand it?
For part iii you simply need to fill in the table according to how $\displaystyle *$ and $\displaystyle \circ$ are defined. For example take the third row of the composition table. In the fourth column you need to enter $\displaystyle 3\circ 4=3*4 (\mod 6)$. Now we have $\displaystyle 3*4=3+4+1=8$ and the remainder of 8 when we divide it by 6 is 2. So the entry in the third row and fourth column of the table is 2. You can use some of the previous parts of the question to make your life easier too- think about how the commutativity of $\displaystyle *$ effects the elements of G under $\displaystyle \circ$
For part iv you need to work out which element is the identity element of G. Then for each $\displaystyle a\in G$ there will exist an integer $\displaystyle m$ such that $\displaystyle a^m$ is the identity element of G. You need to find $\displaystyle m$ for each element $\displaystyle a\in G $ by repeatedly composing $\displaystyle a$ with itself until you get to the identity element.
Once you know the order of each element finding all subgroups of G should be straightforward. Don't forget to use Lagrange- the order of any subgroup divides the order of the group.