# Abstract Algebra, Group Question

## Homework Statement

(a) Suppose a belongs to a group and lal=5. Prove that C(a)=C(a3).

(b) Find an element a from some group such that lal=6 and C(a)≠C(a3).

## 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(a3), 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$\in$G: ga=ag). C(a3) = {g$\in$G: gaa2=a2ag}.

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.

## The Attempt at a Solution

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