A semigroup is a nonempty set with a binary operation on that is associative. To be a group, also needs an identity element and the existence of a two-sided inverse for each element.
Hungerford lists the following propositions that give conditions on when a semigroup is a group:
Hungerford proves Proposition 1 and leaves as an exercise to show Proposition 2 holds from Proposition 1. This is not as simple as it may first appear. Having solutions of immediately gives elements that are left and right identities for particular elements of , but we need all these identities to be equal to each other to satisfy the conditions of proposition 1. The second condition is achieved trivially.
Proof. Let . Find solutions such that . Then find solutions such that and . Call this two-sided identity element of , . Note that So is idempotent. Let and find solutions to . ◻
Here is a different characterization of a group:
Proof. For , denote by the unique element in such that . Then, Therefore, , so that . Also note that must be idempotent because For an idempotent element, so that . Therefore, Implying . Since was arbitrary, this means that is a left identity in . With a symmetric argument, we can prove that is also a right identity. Since was arbitrary, all idempotents are equal to each other. In particular, is an idempotent, hence is an inverse of . ◻