In abstract algebra, 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). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel.
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 very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.
Semigroup and Monoid
Quasigroup and Loop
Map of lattices
Group with operators
Famous quotes containing the word group:
“With a group of bankers I always had the feeling that success was measured by the extent one gave nothing away.”
—Francis Aungier, Pakenham, 7th Earl Longford (b. 1905)