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Multiplicative Group Of Integers Modulo NIn mathematics, the multiplicative group of integers modulo n is the group defined by multiplication of the units (that is, the numbers relatively prime to ) in the ring for a given integer . It is often denoted . The order of the group is given by Euler's totient function. Thus for prime, the order of the group is . This group has many applications in number theory and cryptography. In particular, by finding the size of the group, one can determine if is prime: is prime if and only if the size of the group is . See primality test. The multiplicative group is a cyclic group if and only if , , , or for some odd prime and some . For all other cases, the 2-torsion subgroup is not cyclic (i.e. has a quotient that is a Klein four-group). Using the Chinese remainder theorem, once we determine the structure of the group for prime powers, we can determine the structure of the group for all . By the above, the group is cyclic for odd prime powers. For , the structure of the group is .
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