[15]. is the 2-torsion subgroup (i.e., the square of each element is 1), so is not cyclic. The powers of 3, are a subgroup of order 4, as are the powers of 5, Thus The pattern shown by 8 and 16 holds for higher powers 2k, : is the 2-torsion subgroup (so is not cyclic) and the powers of 3 are a cyclic subgroup of order , so General composite numbers By the fundamental theorem of finite abelian groups, the group is isomorphic to a direct product of cyclic groups of prime power orders. More specifically, the Chinese remainder theorem says that if then the ring is the direct product of the rings corresponding to each of its prime power factors: Similarly, the group of units is the direct product of the groups corresponding to each of the prime power factors: For each odd prime power the corresponding factor is the cyclic group of order , which may further factor into cyclic groups of prime-power orders. For powers of 2 the factor is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above. The order of the group is the product of the orders of the cyclic groups in the direct product. The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function . In other words, is the smallest number such that for each a coprime to n, holds. It divides and is equal to it if and only if the group is cyclic. Subgroup of false witnesses If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power , are congruent to 1 modulo n. (Because the residue 1 when raised to any power is congruent to 1 modulo n, the set of such elements is nonempty.) One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the