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
. ◻