Check Digits and the Pentagon

Abstract

Almost all institutions that rely in a serious way on serial numbers use a check digit scheme to enhance the number and maximize the chance that a computer can detect an error when the number is input. The United States Postal Service, UPS, FedEx, airlines, credit card companies, grocery stores, blood banks, money banks, and driver’s license bureaus. However, all these institutions use imperfect schemes based on normal arithmetic. A perfect scheme, in a sense to be described in §16.2, can be defined if one uses the group of symmetries of a pentagon (a noncommutative group of order 10) to code the 10 digits. In this chapter we will show how the built-in NonCommutativeMultiply operation can be used to do some group theory.