On the existence of super P-groups

A finite group (G, ·) is said to be sequenceable if its elements can be arranged in a sequence a₀ = e, a₁, a₂,…, a_n?1 in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?1 = a₀a₁a₂ ··· a_n?1 are all distinct (and consequently are the elements of G in a new order). It is said to be R-sequenceable if its elements can be ordered in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?2 = a₀a₁a₂ ··· a_n?2 are all different and so that b_n?1 = a₀a₁a₂ ··· a_n?1 = b₀ = e. (in the first case, the ordering a₀,a₁,…,a_n?1 of the elements is said to be a sequencing of G and, in the second case, an R-sequencing of G.) It is a super P-group if every element of one particular coset hG′ of the derived group can be expressed as the product of the n elements of G in such a way that the orderings of the elements in these products are sequencings of G with the exception that, in the case that h = e, the element e of G′ must be expressed as a product of the n elements of G which forms an R-sequencing of G. It is proved that every non-Abelian group of order pq such that p has 2 as a primitive root, where p and q are distinct odd primes with p < q, is a super P-group. Also provided is evidence for the conjectures that all Abelian groups and all dihedral groups of doubly even order (except those of orders 4 and 8) are super P-groups.