TITLE: Why is the identity element included in the definition of a group?
QUESTION [5 upvotes]: Why does definition of a group $G$ always include the requirement that there is identity element $e \in G$ such that for all $a \in G$, $ae=ea=a$, when I prove the existence of such an identity with only the inverse element $a$?

REPLY [12 votes]: You can't define the notion of an inverse element unless you know about an identity element - so you can't prove the existence of an identity element with an argument using inverses.
An argument using inverses might be useful if you want to prove that a subset of a group is a subgroup. Then knowing that the subset was closed under multiplication and contained the inverses of its elements would guarantee that it contained the identity.