Wikipedia definition of a Group:
In mathematics, a group is a set equipped with a binary operation that
combines any two elements to form a third element in such a way that
four conditions called group axioms are satisfied, namely closure,
associativity, identity, and invertibility.
If a set is equipped with a binary operation doesn't it automatically satisfy the closure property? A binary operation on a set $S$ is a mapping of the elements of the Cartesian product $S \times S$ to $S$, so any binary operation on two elements will give an element belonging to the same set.
Now we can have a set equipped with a binary operation where it is not associative example $(\Bbb R,-)$, so it makes sense to specify that it needs to satisfy associativity in order to be called a group.
My question is why do we need to specify the need for satisfying the closure property when we know that it will always be satisfied?
Best Answer
Yes.
A binary operation on a set $S$ is indeed a function from $S\times S$ to $S$.
However, pedagogically, it is helpful to leave in closure as an axiom. It's often the first thing to check for a candidate group whether or not one actually has a binary operation.
It's worth mentioning that there are other, equivalent ways of defining a group, some of which rely on a single axiom.