Computers in Group Theory

Abstract

Much of finite group theory is arithmetical in nature and thus lends itself to computer analysis. In this paper, we describe some computer related projects in group theory which have been done by undergraduates at the University of Minnesota, Duluth, during the past two years. We believe that projects of this kind provide valuable learning experiences for students. The projects we discuss concern two very important classes of finite groups Abelian groups and simple groups. A simple group is a nonabelian group whose only normal subgroups are the identity and the group itself. Thus, as far as the existence of normal subgroups is concerned, Abelian groups and simple groups are at opposite extremes since every subgroup of an Abelian group is normal, while no subgroup (excluding the two trivial cases) of a simple group is normal. Let's consider Abelian groups first. Utilizing the Fundamental Theorem of finite Abelian groups [13, Sec. 2.14] a program was written (all programs were written in Fortran) to determine the isomorphism classes of all Abelian groups of a given order. For example, for the integer 133128 the computer prints out the twelve isomorphism classes of groups of this order as direct products of cyclic groups such as: