A semigroup with only one idempotent element is a group.
This statement is false. Let $A$ be the set of all non-negative integers. For a counterexample, I take $(A,+)$. In the semigroup $(A,+)$, there is only one idempotent element. That is, $0$ since $0+0=0$. In particular, $0$ is the identity element in $(A,+)$. But, $(A,+)$ does not form a group since it does not contain an inverse for all $x \in A$. That is, there is no $y \in A$ such that for all $x \in A, x+y=0$.
Is the counterexample above true?