Note on Hadamard groups and difference sets

A representation theoretical characterization of an Hadamard subset is given. §l. Introduction. A finite group G of order 2n is called an Hadamard group if G contains an n-subset D and an element e* such that (1) D and De*,are disjoint, (2) D and Da intersect exactly in n/2 elements for any element a of G distinct from e* and the identity element e of G, and (3) Da and {b, be*} intersect exactly in one element for any elements a and b of G. The subset D will be called an Hadamard subset corresponding to e*. We consider the group ring of G over the field of complex numbers. If S is a subset of G, then S also denotes the sum of elements of S. Now (1) and (2) together will be expressed as (4) D-1D = ne + (n/2)(G-e-e*) We have shown in (2, Proposition 1) that e* is a central in-volution. For the basic facts on the representations of finite groups the reader is referred to our reference (1). Then we have