Groups

A group is a tuple \((\mathbb{G},\times, 1)\) where \(\mathbb{G}\) is a set of elements, \(\times\) is a binary operator on \(\mathbb{G}\), and \(1\in \mathbb{G}\) is the identity. \(\mathbb{G}\) must have the following properties:

1. *Closure.* \(\times\) is closed. For all \(a\), \(b\in \mathbb{G}\), \((a\times b) \in \mathbb{G}\).
2. Associativity. \(\times\) is associative. For all \(a\), \(b\), \(c\in \mathbb{G}\), \((a\times b)\times c = a\times (b\times c)\).
3. Identity. For all \(a\in \mathbb{G}\), \(a\times 1 = 1\times a = a\).
4. Inverse. For each \(a\in \mathbb{G}\) there is an element \(a^{-1}\in \mathbb{G}\) such that \(a\times a^{-1} = a^{-1}\times a = 1\).