A Note on the Non-existence of Regular Bases for Finite Groups

Abstract

(given by (g1,... , gh) '1 .. gh for gi c S) is surjective and r-to-1 for some integer r. Of course, the integers IS!, IGI, h, and r are necessarily related by SIh = rIGI. A regular h-basis is trivial if S = G, in which case h can be any integer and r = IG h 1. In particular, when r = 1 (the unary case), the only examples for finite groups are h = 1 and S = G; when h > 1 the only example is G = S = {1}. A proof that non-trivial regular h-bases do not exist in finite groups, at least in some cases, was provided by D. Dimovski [3] who gives an example of an infinite group with a non-trivial unary 2-basis. Making explicit use of matrices in the regular representation of a finite group, he establishes that non-trivial regular 2-bases do not exist for any value of r > 1, and then uses this to prove that non-trivial unary h-bases do not exist in any finite group for any h ? 2. This took care of the cases of the boundary values for the parameters h and r. More recently, M. Cushman [2] found an elementary non-representation theoretic proof of this result in the unary case. The elementary argument in [1] is an extension of Cushman's approach, and covers all cases of h ? 2 and r ? 1. Although the argument in [1] is relatively short, and the only machinery required is Fermat's Little Theorem, we believe that for those who have already studied the rudiments of character theory the easiest approach of all is the argument given below (which is different from Dimovski's argument). Even though the proof does rely on character theory, the actual machinery needed (Second Orthogonality) lies at the very surface of the subject matter. It also is complete (covering all cases for h and r) and we believe it illuminates the power of the subject matter. Before beginning the proof, we review some essentials from character theory. A representation of a group G is any homomorphism W: G -* GLn(?) from G to the group of nonsingular n x n matrices over the complex numbers; n is the degree of the representation S. The associated character of (also referred to as the character afforded by -W) is the function X: G -> C obtained by composing with the trace map: