dedicated to professor helmut wielandt on the occasion of his 90th birthday Let 2n be the set of -1 1 sequences of length 2n Then the purpose of the present paper is to introduce the concept of L-structure to elements of 2n and to rewrite the necessary and sufficient conditions for a pair of elements of 2n to be associated in terms of L-structure so that the condition becomes more visible. We hope that this makes it a little easier to construct an associated pair. ยฉ 2000 Academic Press 1. PRELIMINARIES Let G n be a dicyclic group of order 8n presented by G n = a b a 4n = e a 2n = b 2 and b -1 ab = a -1 where e is the identity of G n and n is odd. G n contains only one involution a 2n = b 2 , and it is denoted by e * . Let T be a transversal of G n with respect to e * . If T satisfies the property that T โฉ Tx = 2n 1 for every element of G n outside e * , T is called an Hadamard subset of G n and G n is called a Hadamard group. If G n is an Hadamard group of order 8n then a Hadamard matrix of order 4n will be constructed. For this see [2]. Now any T can be rewritten as T = A + bB where A and B are nsubsets of a . Further an element x in 1 may be restricted to an element 651