Abstract theory of inversion of finite series
Louis Weisner · Transactions of the American Mathematical Society · 1935
LOUIS WEISNER1. Introduction.The summation of a number-theoretic function/(w) over the divisors of n, and the inversion of a series of this type by means of Dedekind's inversion formula, occupy a prominent place in the elementary theory of numbers.fA similar inversion formula is valid in any system whose elements are commutative with respect to a multiplication operation with respect to which a unique factorization law holds, if every element has only a finite number of divisors: for example, primary polynomials in a field, and ideals of an algebraic field.There are, however, systems for which a divisor relation may be properly defined, but for which no unique factorization law holds, and, indeed, in which no rule of multiplication may be defined, as the concept of a divisor is abstractly independent of that of multiplication.For a system of this character the extension of Dedekind's inversion formula is not obvious.An important example is the class of all subgroups of a finite group, with "divisor" defined to mean "subgroup."The problem suggested by Dedekind's inversion formula may be stated as follows: Suppose we are given two group-theoretic functions a(G) and ß(G), such that