Subgroups of Groups of Units mod n

Abstract

The set of all positive integers less than n and relatively prime to n with multiplication mod n is a group denoted U(n). These groups are useful in algebra, number theory and computer science. We are interested in studying the structure of certain subgroups of U(n). As part of their 1980’s paper titled Factoring Groups of Integers Modulo n Gallian and Rusin determined the structure of U(n) and U_s (n) for n=st where gcd(s,t)=1 and U_s (n)={x∈U(n)┤|x (mod s)=1}. We extend this definition to U_k (n) where k is any positive integer and not necessarily a divisor of n. Moreover for a subgroup H of U(n) and an integer k we define: U_(k,H) (n)={x∈U(n)┤|x (mod k)∈H}. We find the structure of these subgroups and the factor group U(n)/U_k (n) in terms of an external direct product of cyclic groups. Our methods also determine group elements of U(n) that form a subgroup with a desired structure. We then shift our attention to the class of subgroups defined as: U(n)^((k))={x^k ┤| x∈U(n)}. We fully classify subgroups of this form and their factor groups. They are useful in finding Sylow p-subgroups of U(n) groups. We also prove some general results about U(n) groups including when the order of U(n) is a power of a prime. Finally we give a simple proof that every finite Abelian group is isomorphic to a subgroup of a U-group.