Quaternionic Starters

Nilpotent 1-factorizations of the complete graph

“…; ð2 m À 1Þ hold. Furthermore, the element 2 mÀ1 a is the unique involution in G. If the group H is trivial, the result follows from Proposition 4 of [2]. Suppose H is non-trivial and consider the group G È H together with the subgroup hai È H. This subgroup has index 2 in G È H and a starter in hai È H was found in Proposition 2.…”

“…The previous Propositions 1, 5, and 6, together with Theorem 2, give a generalization of the main results of [2] and [4]. More precisely, we prove the following Theorem:…”

“…It is also affirmative if G is abelian, and it is not cyclic of order 2n ¼ 2 t > 4, (Buratti [4]), and if G is dihedral (Bonisoli and Labbate [1]). The answer is affirmative for each arbitrary finite non abelian 2-group admitting a cyclic subgroup of index 2 (Bonisoli and Rinaldi [2]). Furthermore, a positive answer is given for some other classes of groups in [3].…”

“…Apart from the dihedral groups and the generalized quaternion groups, there are two more isomorphism types of 2-groups with this property [9, Satz 14.9, p. 90-91]. In both cases, a starter satisfying the conditions of Proposition 1 can be found, see [2]. Proposition 2.…”

“…There are four isomorphism types of such groups [9, Satz 14.9, p. 90-91]; more precisely: the dihedral groups, the generalized quaternion groups, the semidihedral groups, and one more class. For the last two classes, a starter in P which satisfies the conditions of Proposition 1 can be found, see [2], then a starter in G exists. If P is either a dihedral group or a generalized quaternion group, the existence of a starter in G follows, respectively, from Propositions 5 and 6.…”