Combinatorial Problems in Finite Abelian Groups

Publisher Summary This chapter presents combinatorial problems in finite Abelian groups. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. Abelian groups generalize the arithmetic of addition of integers. The concept of an Abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-Abelian counterparts, and finite Abelian groups are also well understood. Every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated Abelian groups when G has zero rank.