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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?

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  • $\begingroup$ Yes, but sometimes you need to check that a supposed operation really is an operation (i.e., is closed). $\endgroup$ – Randall Apr 12 at 17:12
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    $\begingroup$ consider a subgroup $\endgroup$ – J. W. Tanner Apr 12 at 17:14
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If a set is equipped with a binary operation doesn't it automatically satisfy the closure property?

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.

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