The Structure of Finite Abelian Groups
Steven J. Rosenberg · 2021
Setwise products are defined in arbitrary groups, using additive notation for abelian groups; attention is given to the case when a product of subgroups is a subgroup. A formula is proved for the cardinality of a product of two finite subgroups. Conditions are developed to determine when the natural map from a direct sum of subgroups is an isomorphism. Finite p-groups are defined, and finite abelian groups are shown, using the Chinese Remainder Theorem, to decompose into maximal p-subgroups. Finite abelian p-groups come under scrutiny next; the exponent of a group is defined, and the vector space structure of elementary abelian p-groups is exploited to give their direct sum decomposition. Connections between direct sum decompositions and bases of vector spaces are explicitly drawn, preparing the reader for a later treatment of the decomposition of finitely generated modules over a principal ideal domain. The idea of a characteristic subgroup is introduced, and examples given. After this ample preparation, the Fundamental Theorem of Finite Abelian Groups is stated and proved, including the uniqueness of the exponent sequences. The chapter concludes with a section of exercises.