Every finite solvable group with a unique element of order two, except the quaternion group, has a symmetric sequencing
✍ Scribed by B. A. Anderson; E. C. Ihrig
- Publisher
- John Wiley and Sons
- Year
- 1993
- Tongue
- English
- Weight
- 591 KB
- Volume
- 1
- Category
- Article
- ISSN
- 1063-8539
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✦ Synopsis
Abstract
All 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.