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Chinese Journal of Mathematics
Volume 2014 (2014), Article ID 950572, 4 pages
http://dx.doi.org/10.1155/2014/950572
Research Article

Splitting Groups with Basis Property

Al Imam Muhammad Ibn Saud Islamic University, P.O. Box 90189, Riyadh 11613, Saudi Arabia

Received 8 October 2013; Accepted 12 November 2013; Published 6 February 2014

Academic Editors: E. Bannai and H. You

Copyright © 2014 Abdullah Aljouiee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A finite group is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group is said to have the basis property if, for each subgroup , has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non- -group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any -group has basis property. I demonstrate some types of -groups which are splitting as well.