Application of Gaschütz' Theorem to relative difference sets in non-abelian groups
✍ Scribed by John C. Galati
- Publisher
- John Wiley and Sons
- Year
- 2003
- Tongue
- English
- Weight
- 80 KB
- Volume
- 11
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
Let G be a finite group other than ℤ~4~ and suppose that G contains a semiregular relative difference set (RDS) relative to a central subgroup U. We apply Gaschütz' Theorem from finite group theory to show that if G/U has cyclic Sylow subgroups for each prime divisor of |U|, then G splits over U. A corollary of this result is that a finite group (other than ℤ~4~) in which all Sylow subgroups are cyclic cannot contain a central semiregular RDS. We also include an example, originally discovered by D.L. Flannery, which shows that our main theorem is not true in general when U is a (not necessarily central) abelian normal subgroup of G. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 307–311, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10041