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Determine whether (N,*) is a group or not. N={0,1,2,3,...} and

n*m= n+m if n,m even, -(n+m)-2 if n,m odd, and m-n-1 if n odd, m even.

I know that the properties of a group are associativity, existence of an identity, and inversiblity. If I consider the case where n,m are both even, clearly the third condition is not satisfied since there does not exist an additive inverse in N. Is this enough to conclude that it is not a group, or do I need to consider the other cases?