Theoretical approach of a finite abelian groups

It is hopeless to classify all infinite abelian groups, but a good criterion that leads to an interesting classification is that of finite abelian groups. A finite Abelian group G is a p-group with p ∈ N a prime then every element of G has order a power of p. the order of a finite p-group must be a power of p .A finite abelian group is a group satisfying the following equivalent conditions. It is isomorphic to a direct product of finitely many finite cyclic groups. It is isomorphic to a direct product of abelian groups of prime power order and it is also isomorphic to a direct product of cyclic groups of prime power order. 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 (the axiom of commutativity). They are named after Niels Henrik Abel. An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.