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## Homework Statement

(a) Suppose a belongs to a group and lal=5. Prove that C(a)=C(a

^{3}).

(b) Find an element a from some group such that lal=6 and C(a)≠C(a

^{3}).

## Homework Equations

## The Attempt at a Solution

For (a) I know I need to show that every element in the set C(a) is also an element in the set C(a

^{3}), but am having trouble proving it.

I believe that every element from a group should commute with itself. So if the group were the rotations of a triangle with composition as the operation, a 120 degree rotation followed by a 120 degree rotation is commutative. It would make sense that all powers of an element should commute with each other. So C(a) is the set {g[itex]\in[/itex]G: ga=ag). C(a

^{3}) = {g[itex]\in[/itex]G: gaa

^{2}=a

^{2}ag}.

I feel I am on the right track, just can't seem to put the final nail in.

For part (b), not grasping this part. I saw someone at the math center using a hexagon and rotations, but it seems all rotations commute.

Trying to work out an example using modular arithmetic too but coming up with nothing. Any guidance would be greatly appreciated.