let s be a semigroup. prove that s is a group if and only if the following conditions hold:
1. s has a left identity—there exists an element e ∈ s such that ea = a for all a ∈ s;
2. each element of s has a left inverse—for each a ∈ s, there exists an element a -1 ∈ s such that a -1a = e.