Why is closure omitted in some group definitions? 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?
 A: Without knowing the full context, I will guess that such texts defined $*$ to be a 'binary operation,' or some other definition that somehow encompasses closure within it but is slightly hidden. I have read a text that also assumed associativity of the binary operation, so that all that was needed was the existence of the identity and the inverse. But, defining a group with a binary operation or a group with an arbitrary operation that is closed and associative is more or less the same thing.
At the end of the day, the group operation is closed and associative, regardless of how explicit or implicit the book makes it.
I know I typed this up without knowing the exact context, but is that right? Did that book use some pre-defined or specified operator?
A: 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.
A: When three axioms are given (associativity, existence of an identity, existence of inverses), these are axioms that apply to a set $G$ with a binary operation.  A binary operation is by definition a function from $G\times G$ to $G$, meaning that for each ordered pair $(a,b)$ of elements of $G$, there is a unique element (denoted $a*b$ or $ab$ or some other way) of $G$ determined by the operation.  Thus, the "closure axiom" is incorporated into the definition of a binary operation on a set.
