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.

An approach to simulate basic concepts of mathematics with object-oriented programming

“…It is provable that for each call of the operator "%", the parameters are two ElementOfIntegerSet instances which simulate two elements from the set of integers, and the return type is also ElementOfIntegerSet so that the return value is also an instance of ElementOfIntegerSet simulating an element from the set of integers, which means that this operator "%" method has closure on the set of all integers. Notice that the operator "%", which is the modular arithmetic, is very important when expanding the conclusions in this paper to further algebra concepts such as groups [10] so that it should be carefully overloaded. The example above is just one of the simplest realizations and may not be precise in future works.…”