R-sequenceability and R*-sequenceability of abelian 2-groups

A group of order n is said to be R-sequenceable if the nonidentity elements of the group can be listed in a sequence a,, a,, . . . , a,_ 1 such that the quotients a; ‘az, a; ’ a3, . , a;?,a,_ r, a.-_‘,a, are distinct. An abelian group is R*-sequenceable if it has an R-sequencing a,, a,, . . , a,, _ 1 such that a,_ ra, + r = ai for some i (subscripts are read modulo n 1). Friedlander, Gordon and Miller (1978) showed that an R*-sequenceable Sylow 2-subgroup is a sufficient condition for a group to be R-sequenceable. In this paper we also show that all noncyclic abelian 2-groups are R*-sequenceable except for Tz x ZT4 and ZTz x ?Xz x T1. A group of order n is said to be R-sequenceable if the nonidentity elements of the group can be listed in a sequence al, a2,. . . , a,_ 1 such that the quotients a; ‘az, a; lu 3, . . . , u;12un_ 1, a.=‘lal are distinct. The concept of R-sequenceability has been around for more than 40 years in one form or another. In 1951 Paige observed that it is a sufficient condition for a group to have a complete mapping. In 1955 Hall and Paige [3] showed that a solvable group has a complete mapping if and only if its Sylow 2-subgroup is either trivial or noncyclic. In 1974 Ringel [S] was led to the concept of R-sequenceability in his solution of the map coloring problem for all compact two-dimensional manifolds except the sphere. In their book [l] D&es and Keedwell used an alternative definition of R-sequenceable and discussed the topic in great depth. They also showed that an abelian group is a super P-group if and only if it is either R-sequenceable or sequenceable. Friedlander et al. [2] showed that the following types of abelian groups are R-sequenceable: cyclic groups of odd order greater than 1; groups of odd order whose Sylow 3-subgroup is cyclic; groups whose orders are relatively prime to 6; elementary abelian p-groups, except the group of order 2; groups of type L$?~ x sdk, k 3 1; groups whose Sylow p-subgroup has the form 27, m > 1 but m # 3; groups G whose Sylow p-subgroup has the form S=.Zz x %,, 0012-365X/94/$07.00