Group covers and partitions : covering and partition numbers

WCU Author/Contributor (non-WCU co-authors, if there are any, appear on document)
Nicholas Sizemore (Creator)
Western Carolina University (WCU )
Web Site:
Tuval Foguel

Abstract: A group cover is a collection of subgroups whose union is the group. A group partition is a group cover in which the elements have trivial pairwise intersection. Previous work has been done to investigate properties related to the covering number of a group - the minimal number of subgroups necessary to forma cover. Here we define analogously the partition number of a group and examine some of its properties, including its relation to the covering number. In particular, we provide a classification of the covering and partition numbers of the dihedral groups. Also presented is a result concerning a minimal partition of the symmetric group on four elements. Further, we utilize GAP to provide some computational methods for studying partition numbers. Code samples are provided to test conditions relevant to partitions and the partition number; specifically, we are able to use GAP to help verify various conjectures about partitions of small finite groups. Included is a naïve and computationally intense method to find the partition number, as well as an attempted refinement designed to deal with the inefficiency of the naïve method.

Additional Information

Language: English
Date: 2013
covering number, dihedral group, gap, group covers, group partitions, partition number
Group theory
Partitions (Mathematics)

Email this document to