text: # Thread: How to prove a monoid is a group.
  
  1. ## How to prove a monoid is a group.
  
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  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?
  
  2. Originally Posted by pikminman
  Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula
  
  $x * y = xy + c(x+y) + c^2 - c$
  
  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?
  A group is a monoid where every element has an inverse, $a*a' = e$ where $e$ is your identity and $a$ your element. So, you just plug in you $a$ and $a'$ and see what $a'$ must be for $a*a' = e$.
  
  In these questions always look out for possible illegal steps, such as dividing by zero. Often most elements will have an inverse, but some wont, and the ones which wont will be the ones where zero is in the botton of your inverse...