Explore: Abelian Groups

quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of

. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups. Every finite abelian group is finitely generated. The

theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic

compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex

topological abelian group, or TAG, is a topological group that is also an abelian group. That is, a TAG is both a group and a topological space, the group operations

has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category

The motivating prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for

a different meaning in the context of elementary abelian groups. A subset {aα} of an abelian group A is linearly independent (over Z) if the only linear

a ∗ b ≠ b ∗ a. This class of groups contrasts with the abelian groups, where all pairs of group elements commute. Non-abelian groups are pervasive in mathematics

algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined