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In mathematics, the multiplicative group of integers modulo n
is the group defined by multiplication of the units in the ring
- <math>\mathbb{Z}/n\mathbb{Z}<math>
for a given integer n > 1. The order of the group is given by Euler's totient function. Where <math>n<math> is prime, the order of the group is <math>n-1<math>.
This group has many applications in number theory and cryptography. Its structure is reduced to that of the prime power case by use of the Chinese remainder theorem.
The multiplicative group is a cyclic group if and only if <math>n = 2<math>, <math>n = 4<math>, <math>n = p^m<math>, or <math>n = 2p^m<math> for some prime <math>p<math> and some <math>m > 0<math>. For all other cases the 2-torsion subgroup is not cyclic (i.e. has a quotient that is a Klein four-group), ruling that out.
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