Exercise 8, § 2

In the notation of Exercise 7, suppose A is finite; show that A contains one and only one idempotent ($§ 1$, no. 4) (observe that if $x^p$ and $x^q$ are idempotents, then $x^p = x^{pq}, x^q = x^{pq}$, hence $x^p = x^q$); if $h = x^p$, the set of $x^n$ for $n \geq p$ is a stable subset D of E such that under the law induced on D, D is a group.