mathematics, 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...

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is...

In abstract algebra, an abelian group ( G , + ) {\displaystyle (G,+)} is called finitely generated if there exist finitely many elements x 1 , … , x s...

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...

mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at...

theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped...

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

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...

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

an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained...

In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces...

specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently...

In mathematics, specifically in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided...

In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same...

In geometry, an abelian Lie group is a Lie group that is an abelian group. A connected abelian real Lie group is isomorphic to R k × ( S 1 ) h {\displaystyle...

algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative...

notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their...

is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear);...

In mathematics, a character group is the group of representations of an abelian group by complex-valued functions. These functions can be thought of as...

groups). If G and H are abelian (i.e., commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group:...

In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced...

mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in...

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also...

In group theory, a dicyclic group (notation Dicn or Q4n, ⟨n,2,2⟩) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of...

said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative...

metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup...

such that the quotient group of the original group by this subgroup is abelian. In other words, G / N {\displaystyle G/N} is abelian if and only if N {\displaystyle...

nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two...

finite p-group is nilpotent. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and...