It can be shown that if formula_139 and formula_140 are equivalent to formula_145 and formula_146 respectively, then the composition of formula_139 and formula_140 is equivalent to the composition of formula_145 and formula_146. It follows that composition induces a well-defined operation on primitive classes of discriminant formula_28, and as mentioned above, Gauss showed these classes form a finite abelian group. The identity class in the group is the unique class containing all forms formula_152, i.e., with first coefficient 1. (It can be shown that all such forms lie in a single class, and the restriction formula_153 implies that there exists such a form of every discriminant.) To invert a class, we take a representative formula_138 and form the class of formula_155. Alternatively, we can form the class of formula_156 since this and formula_155 are equivalent.