Sooji Shin

Theorem.  Let G be a nonempty set with a binary operation and satisfy the following.
(a) Associative rule.  i.e. a(bc)=(ab)c for all a, b, c in G.
(b) Existence of left identity.  i.e.  There exists e in G s.t. ea = a for all a in G.
(c) Existence of left inverse element.  i.e. For all a in G there exists a' in G s.t. a' a = e.
Then G is a group.

Proof.  Denote a'  for the left inverse element of a in G.
First, we see as a lemma that c² = c implies c = e, for

Now let a be an arbitrary element of G. We have to show that a', the left inverse element of a, is the right inverse element and that e, the left identity element, is the right identity element. Observe that
which implies aa' = e by the lemma above. Thus we conclude that a' is the right inverse element of a. Besides, we have
Thus we conclude that e is the right identity element.
 

Corollary.  Let G be a nonempty set with a binary operation and satisfy the following.
(a) Associative rule.  i.e. a(bc)=(ab)c for all a, b, c in G.
(b) Existence of right identity.  i.e.  There exists e in G s.t. ae = a for all a in G.
(c) Existence of right inverse element.  i.e. For all a in G there exists a' in G s.t. aa' = e.
Then G is a group.

Geschrieben von Sooji Shin