#### Abstract

Let G be a finite, nontrivial group. In a paper in 1994, Cohn defined the covering number of a finite group as the minimum number of nontrivial proper subgroups whose union is equal to the whole group. This concept has received considerable attention lately, mainly due to the importance of recent discoveries. In this thesis we study a dual concept to the covering number. We define the intersection number of a finite group as the minimum number of maximal subgroups whose intersection is equal to the Frattini subgroup. Similarly we define the inconjugate intersection number of a finite group as the minimum number of inconjugate maximal subgroups whose intersection is equal to the Frattini subgroup. We study basic properties for these intersection numbers and determine its values for certain types of finite groups. Then we single out some similarities and differences between the covering number and the intersection number in specific types of groups. Finally we point to some directions of future developments and research.

#### Date of publication

Spring 5-14-2019

#### Document Type

Thesis

#### Language

english

#### Persistent identifier

http://hdl.handle.net/10950/1332

#### Committee members

Dr. Kassie Archer, Dr. Scott LaLonde, Dr. Lindsey-Kay Lauderdale, Dr. David Milan

#### Degree

Masters in Mathematics

#### Recommended Citation

Bautista Serrano, Humberto, "ON THE INTERSECTION NUMBER OF FINITE GROUPS" (2019). *Math Theses.* Paper 9.

http://hdl.handle.net/10950/1332