Finite Abelian Groups
Cyclic groups of integers modulo n, Z/nZ, were among the first examples of groups. It turns out that 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.
Read more about this topic: Abelian Group
Famous quotes containing the words finite and/or groups:
“Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are prefabricated in the sense that we dont coin new ones every time we speak.”
—David Lodge (b. 1935)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (18721945)