Letcbe a fixed positive integer, and let * denote the binary operation on the setZof integers defined be the formula

$\displaystyle x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a group?

I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?