Abstract Algebra, Group Question

[srfriggen]

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).

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(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\inG: ga=ag). C(a³) = {g\inG: 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.

 

[Dick]

All rotations do commute with each other. Looking at the symmetries of the hexagon will give you an example for part b). Rotations aren't the only elements of that dihedral group. There are reflections involved as well. What about them? If a is a rotation by 60 degrees, compare C(a) with C(a^3).

Then give some more thought to part a). Knowing all rotations commute doesn't help.