On left Hadamard transversals in non-solvable groups

Let GG be a group of order 4n4n and tt an involution of GG. A 2n2n-subset RR of GG is called a left Hadamard transversal of GG with respect to 〈t〉〈t〉 if G=R〈t〉G=R〈t〉 and R^R(-1)^=nS1^+2nS2^ for some subsets S1S1 and S2S2 of GG. Let HH be a subgroup of GG such that G=[G,G]HG=[G,G]H, t∈Ht∈H, and tG⊄HtG⊄H, where tGtG is the conjugacy class of tt and [G,G][G,G] is the commutator subgroup of GG. In this article, we show that if RR satisfies a condition (*)R≠xR∀x∈G⧹{1}, then RR is a (2n,2,2n,n)(2n,2,2n,n) relative difference set and one can construct a v×vv×v integral matrix BB such that BBT=BTB=(n/2)IBBT=BTB=(n/2)I, where vv is a positive integer determined by HH and tGtG (see Theorem 2.6). Using this we show that there is no left Hadamard transversal RR satisfying (*)(*) in some simple groups.