Group (S, *) is a monoid with inverse elements.

Requirements

1. There's an identity element \(id\): \(a \diamond id = id \diamond a = a\).
2. \(\diamond\) is associative: \((a \diamond b) \diamond c = a \diamond (b \diamond c)\).
3. There's an inverse lement: \(a a^{-1} = a^{-1} a = 1\).