Question

For what values of n, u(n) is a cyclic group

Answer 1

Un={a ∈ Zn ∣ gcd (a,n)=1}


So Un is the group of units in Z / nZ.Write the prime decompositionn=pα11⋯pαrr.n=p1α1⋯prαr.By the Chinese remainder theoremZ/nZ=Z/pα11Z×…×Z/pαrrZZ/nZ=Z/p1α1Z×…×Z/prαrZsoUn=Upα11×…×Upαrr.Un=Up1α1×…×Uprαr.For powers of 22, we haveU2={0}U2={0}and for k≥2k≥2U2k=Z/2Z×Z/2k−2Z.U2k=Z/2Z×Z/2k−2Z.For odd primes pp,Upα=Z/ϕ(pα)Z=Z/pα−1(p−1)Z.Upα=Z/ϕ(pα)Z=Z/pα−1(p−1)Z.So you see now that UnUn is cyclic if and only ifn=2,4,pα,2pαn=2,4,pα,2pαwhere pp is an odd prime.

Answer 2

Un={a∈Zn∣gcd(a,n)=1}
:)