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A binary operation on a set \(A\) is just a function \(\ast: A \times A \rightarrow A\). For any \(a,b \in A\), we write \(\ast(a,b)=a \ast b\), or \( \ast: (a,b) \rightarrow a \ast b\). remark (sidenote: When context is clear, we may abbreviate \(a \ast b\) as \(ab\). )

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 \(a \in G\). remark (sidenote: We remark that the closure axiom is redundant in this definition.