The Rank of a Latin Square associated to an abelian group

Abstract

Let G ≃ ℤ/2ℤ × ℤ/2ℤ × Πi=1 r (ℤ/pi tiℤ) be a finite abelian group, where pi (1 ≤ i ≤ r) are (not necessarily distinct) odd primes. Suppose x = ∑g∈Gxgg ∈ ℤ[G] with {xg : g ∈ G} = {1, . . . , |G|}. Using a result of Carlitz and Moser, we show that |{χ ∈ G* : χ(x) ≠ 0}| ≥ 3 + ∑ φ)pi ti). Consequently, we prove that the rank of any Latin square associated with the group G is at least 3 + ∑ φ(pi ti). This sharpens a result in [2]. Copyright © 2000 by Marcel Dekker, Inc.