#### Publication Date

11-1978

#### Advisor(s) - Committee Chair

Kyle Wallace, Carroll Wells, Wilburn Jones

#### Degree Program

Department of Mathematics and Computer Science

#### Degree Type

Master of Science

#### Abstract

In their paper “Groups as Unions of Proper Subgroups,” published in 1959, Haber and Rosenfeld posed the problem of characterizing groups which can be expressed as the set theoretic union of a finite collection of their proper subgroups. Haber and Rosenfeld established that a group G is the union of three proper subgroups (such a group G shall be called a U_{3}-group) if and only if the Klein 4-group is a homomorphic image of G. Further results concerning U_{3}-groups were obtained in the paper “Groups which are the Union of Three Subgroups” by Bruckhemier, Bryan and Muir (1970), and in the paper “On the Number of Subgroups of Index Two – An Application of Goursat’s Theorem for Groups” by Crawford and Wallace (1975). The purpose of this paper is to provide characterizations for groups that can be expressed as the union of four proper subgroups but that cannot be expressed as the union of fewer than four such subgroups. We shall refer to such groups as U_{4}-group if and only if either the symmetric group on three letters, S_{3}, or the direct product of two cyclic groups each having order 3, Z_{3}xZ_{3}, is a homomorphic image of G.

#### Disciplines

Applied Mathematics | Mathematics | Physical Sciences and Mathematics

#### Recommended Citation

Barrett, Lana, "Groups Expressed as the Set-Theoretic Union of Proper Subgroups" (1978). *Masters Theses & Specialist Projects.* Paper 1879.

http://digitalcommons.wku.edu/theses/1879