Beautiful problem! (It was posted here before).

Define the homomorphism:

as .

Notice that,

.

So, is 1-to-1 (it is not divisible by 3).

By Pigeonhole Principle issurjective.

Now consider,

So,

Since is surjective can represent any element in so,

.

Hence,

the center of (because it commutes with everything).

Now,

---> since .

Q.E.D.

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Try doing the problem after that in Herstein's book