Monoid is a group if
1
a * c = c * a = e, for all a ∈ G and c is also belongs to G
2
a * (c * a) = ( a * c) * a, for all a ∈ G and c is also belongs to G
3
a * c = a + c, for all a ∈ G and c is also belongs to G
4
a * c = a % c, for all a ∈ G and c is also belongs to G