Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$.

Question

Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$.

For the right direction, since $G$ is a group with identity, then there exists $a=e_G$ i.e. $a^2 = a$. Isn't it?

For the left direction, what should I do?