# Binary Operations : Idempotence

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An element e of a monoid M is called an idempotent if e^2 = e. If M is finite, show that some positive power of every element is an idempotent.

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#### Solution Preview

Prove by induction:

e is idempotent, then e^2=e (1)

Assuming when the positive power of e is n=k, then (e^k) ...

#### Solution Summary

Idempotence of powers of elements in a monoid is demonstrated by mathematical induction. The positive power of every element is determined.

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