CHAPTER IX - HOMOMORPHISM GROUPS AND ENDOMORPHISM RINGS

Abstract

This chapter discusses homomorphism groups and endomorphism rings. From the rudiments of the general theory of groups, it is known that the endomorphisms of a group form an associative ring under a natural definition of addition and multiplication provided the group is commutative. The chapter focuses on the character group. The chapter explains how the isomorphy of two endomorphism rings implies the isomorphy of the two p-groups themselves. There is a strong connection between the structure of a group and the structure of its endomorphism ring. The homomorphism group of a torsion group U into some group V is the complete direct sum of the homomorphism groups of the p-components of U into the corresponding p-components of the maximal torsion subgroup of V.