TITLE: Why is closure omitted in some group definitions?
QUESTION [13 upvotes]: In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?

REPLY [3 votes]: Stating closure explicitly allows the definition of a subgroup to simply be a subset that is a group when you restrict the group operation to it.
The point here is that when you restrict the operation of $G$ to a subset $H$, you get a map $H \times H \to G$, and so closure for $H$ means this map is actually $H \times H \to H$.
Assuming closure as implicit in the definition of binary operation is fine, but it does make the definition of subgroup slightly more wordy.
Of course, this is a matter of taste and different writers choose different definitions according to their taste.