Then "N" is normal in "G" and the factor group "G"/"N" has the cosets "N", (1+"n")"N", (1+"n")2N, ..., (1+"n")"n"−1N. The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of "G" without knowing the factorization of "n". The quotient group is isomorphic to the trivial group (the group with one element), and is isomorphic to "G". The order of , by definition the number of elements, is equal to , the index of "N" in "G". If "G" is finite, the index is also equal to the order of "G" divided by the order of "N". The set may be finite, although both "G" and "N" are infinite (for example, ). There is a "natural" surjective group homomorphism , sending each element "g" of "G" to the coset of "N" to which "g" belongs, that is: . The mapping "π" is sometimes called the "canonical projection of G onto ". Its kernel is "N". There is a bijective correspondence between the subgroups of "G" that contain "N" and the subgroups of ; if "H" is a subgroup of "G" containing "N", then the corresponding subgroup of is "π"("H"). This correspondence holds for normal subgroups of "G" and as well, and is formalized in the lattice theorem. Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems. If "G" is abelian, nilpotent, solvable, cyclic or finitely generated, then so is . If "H" is a subgroup in a finite group "G", and the order of "H" is one half of the order of "G", then "H" is guaranteed to be a normal subgroup, so exists and is isomorphic to "C"2. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if "p" is the smallest prime number dividing the order of a finite group, "G", then if has order "p", "H" must be a normal subgroup of "G".