A group is a set \(G\) with a [[binary-operation|binary operation]] \(*\) such that the following conditions are satisfied: 1. \(*\) is associative; 2. \(*\) has an identity; and 3. Every element of \(G\) has an inverse.
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We say that \(G\) is a group “under” or “with respect to” the binary operation \(*\).
A group whose binary operation is commutative is called commutative or abelian.
We write \(|S|\) for the number of elements of the (finite) set \(S\); if \(G\) is a group, we call \(|G|\) the group order.