Construction of multiplicative groups in modular arithmetic

Abstract

There are multiplicative groups in modular arithmetic. It is known that when n is a positive integer, the set of all positive integers less than and relative prime to n is a group under multiplication modulo n. Some authors have studied multiplicative groups in modular arithmetic, and many of these groups have been constructed. In this paper we review some of the constructions, including constructions using elements of a geometric sequence and elements of an arithmetic sequence. Some of the constructions are extensions of the existing groups, and some others are new constructions. We also show that it is possible to find other new constructions.