Group
A group is a non-empty set \(G\) together with a binary operation \(\ast: G \times G \rightarrow G\) such that the following hold:
(Closure) For every \(a,b \in G\), the product \(a \ast b \in G\).
(Associativity) The binary operation \(\ast: G \times G \rightarrow G\) is associative .
(Identity) \(G\) has an identity element .
(Inverse) There exists an inverse element for every element \(a \in G\).
(sidenote: We remark that the closure axiom is redundant in this definition. Indeed, by definition of a binary operation we are guaranteed that: \(\ast(G \times G) \subseteq G\). )