A.3 Groups Consider a prime number p . A group is usually identified with its underlying set, unless the group operation is not clear from context. G is a group such that the set H is a subset of G . This means that every element in G is of the form g k for some integer k .
All cyclic groups are abelian.
The set Z of integers is a cyclic group under addition with generator 1. However, the set of nonzero integers is not a group under multiplication. The sets Q , R , and C of rational, real, and complex numbers are all abelian groups under addition. The set Z n is a cyclic group under addition. Namely, the product of a and b is equal to 0 modulo n , which implies that the set is not even closed under multiplication.