ABSTRACT

G H G×H harmonious harmonious, of odd order harmonious sym. harmonious, symmetrically harmonious, symmetrically harmonious of odd order of odd order R∗-sequenceable symmetrically harmonious R∗-sequenceable, with the possible

exception of |G| odd and 3||H| R∗-sequenceable, abelian, |H| odd R∗-sequenceable of even order sym. sequenceable abelian, (|G|, |H|) = 1 symmetrically sequenceable

6.61 A 2-sequencing of a group G of order n is a sequence a0 = e, a1, a2, . . . , an−1 of elements of G in which g ∈ G appears exactly once if g = g−1, and |{i | ai ∈ {g, g−1}, i = 0, . . . , n − 1}| = 2 if g 6= g−1, such that all of the partial products b0 = a0 = e, b1 = a0a1, b2 = a0a1a2 , . . . , bn−1 = a0a1a2 · · ·an−1 are distinct. A group is 2-sequenceable if it possesses a 2-sequencing.