Every finite solvable group with a unique element of order two, except the quaternion group, has a symmetric sequencing

Abstract

AbstractAll finite solvable groups that have symmetric sequencings are characterized. Let G be a finite solvable group. It is shown that G has a symmetric sequencing if and only if G has a unique element of order two and is not the quaternion group. All finite groups with a unique element of order two such that the order of the group is not divisible by three are solvable and thus, except for the quaternion group, have symmetric sequencings. A crucial step used in the proof of these facts is a construction showing that if a finite group H has a normal subgroup C of odd order such that H/C admits a 2‐sequencing, then H admits a 2‐sequencing. The results of this article can be viewed as generalizing a theorem of Gordon about Abelian groups and as extending the idea of a starter, suitably modified, to a large class of groups of even order by showing the existence of the required object. © 1993 John Wiley & Sons, Inc.