On left Hadamard transversals in non-solvable groups

Let G be a group of order 4n and t an involution of G. A 2n-subset R of G is called a left Hadamard transversal of G with respect to 〈t〉 if G=R〈t〉 and for some subsets S₁ and S₂ of G. Let H be a subgroup of G such that G=[G,G]H, t∈H, and t^G⊄H, where t^G is the conjugacy class of t and [G,G] is the commutator subgroup of G. In this article, we show that if R satisfies a condition , then R is a (2n,2,2n,n) relative difference set and one can construct a v×v integral matrix B such that BB^T=B^TB=(n/2)I, where v is a positive integer determined by H and t^G (see Theorem 2.6). Using this we show that there is no left Hadamard transversal R satisfying (*) in some simple groups.