Basic Theory of Additive Abelian Groups

Abstract

AbstractThe theory of abelian groups, also called ℤ-modules, is developed from scratch. Subgroups and homomorphisms are renamed submodules and ℤ-linear mappings to emphasise the analogy with vector spaces. Group isomorphisms, cyclic groups and their isomorphism types are introduced. Special attention is given to quotient groups G/K. Direct sums are studied, the Chinese remainder theorem being regarded as a direct sum of rings. The Euler ϕ-function is multiplicative. Independent submodules and decompositions. The first isomorphism theorem for ℤ-modules and the corresponding lattices of submodules. R-modules where R is a commutative ring (the reader is expected to be familiar with rings, fields, matrices and abstract linear algebra including bases). Free R-modules and rank, R-bases. R-linear mappings, isomorphic R-modules, submodules and quotient modules. The first isomorphism theorem for R-modules.KeywordsAbelian GroupCyclic GroupAdditive GroupQuotient GroupCyclic SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.