prove:U(p)is isomorphic to Z(p-1),where p is prime.Z(p-1)is the additive group of modulo p-1.U(p) means all the positive intergers that is relatively prime to p.
Thanks a lot!!
In fact this is a consequence of a more general theorem.
A finite subgroup of the multiplicative group of a field is cyclic. In particular, if F is a finite field, then the multiplicative group of nonzero elements of F is a cyclic group.
By The Fundamental Theorem, the finite subgroup can be written as a direct product of cyclic groups
Where .
In general, if G is a cyclic group and , then G contains precisely d elements of order dividing d. Since divides the order of each of the cyclic groups in the direct product, it follows that each direct factor contains elements of order dividing . If k were greater than 1, there would therefore be a total of more than such elements. But then there would be more than roots of the polynomial in the field F, which contradicts the fact that a polynomial of degree n can have at most n roots. Thus k=1 and the group is cyclic.
Notice is a finite field, so your question is a trivial corollary of the above theorem.