Editing Finite group (section)

===Finite abelian groups===
{{main|Finite abelian group}}
An '''[[abelian group]]''', also called a '''commutative group''', is a [[group (mathematics)|group]] in which the result of applying the group [[Operation (mathematics)|operation]] to two group elements does not depend on their order (the axiom of [[commutativity]]). They are named after [[Niels Henrik Abel]].<ref>{{harvnb|Jacobson|2009|p=41}}</ref>

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]].