Divisibility Among Power GCD Matrices and Power LCM Matrices

For any integers x and y , let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y , respectively. Let $a,b$ and n be positive integers, and let S=₁, … , xₙ\ be a set of n distinct positive integers. We denote by (Sᵃ) and [Sᵃ] the n× n matrices having the a th power of (xᵢ,xⱼ) and [xᵢ,xⱼ] , respectively, as the $(i,j)$ -entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S ), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as (S) [S] ) in the ring Mₙ( Z) of n× n matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that (Sᵃ) (Sᵇ) , (Sᵃ) [Sᵇ] and [Sᵃ] [Sᵇ] in the ring Mₙ( Z) when a b and S is a divisor chain (namely, there is a permutation σ of order n such that xσ (1) ⋯ xσ (n) ). In this paper, we show that if a b and S is factor closed, then (Sᵃ) (Sᵇ) , (Sᵃ) [Sᵇ] and [Sᵃ] [Sᵇ] in the ring Mₙ( Z) . The proof is algebraic and p -adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.