An extension theorem for terraces. (Q1953523)

Let \(\mathbf a=(a_1,a_2,\ldots,a_n)\) be an arrangement of all of the elements of group \(G\), set \(\mathbf b=(b_1,b_2,\ldots,b_{n-1})\) by \(b_i=a_i^{-1}a_{i+1}\). If \(\mathbf b\) contains each involution of \(G\) exactly once and exactly two occurrences from each set \(\{g,g^{-1}:g^2\neq e\}\), then \(\mathbf a\) is a terrace for \(G\) and \(\mathbf b\) is its associated 2-sequencing. In 1984, Bailey introduced terraces for arbitrary finite groups in connection with the construction of quasi-complete Latin squares and he conjectured that all groups except the non-cyclic elementary Abelian 2-groups were terraced (known as Bailey's Conjecture). In this paper, the authors present a more general version of terraces for groups with a central subgroup isomorphic to the Klein \(4\)-group and construct extendable terraces for the following three non-Abelian groups of order \(8t\) with \(t\geq 2\): the dihedral group \(D_{8t}\), the semidihedral group \(S_{8t}\) and the group \(M_{8t}=\langle u,v:u^{4t}=e=v^2,\;vu=u^{2t+1}v\rangle\), which makes a step forward for the proof of Bailey's Conjecture.