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...

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

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...

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

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...

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...

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 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...

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

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...

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

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...

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

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...

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:...

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...

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 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...

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...

cyclic group. An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements...

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 mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all...

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...

\mathbb {Z} /(p-1)\mathbb {Z} } In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by...

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...

abelian group is an abelian group in which every element has finite order. For example, the torsion subgroup of an abelian group is a torsion abelian...

isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative)...

multiplicative group of positive real numbers R + {\displaystyle \mathbb {R} ^{+}} is an abelian group with 1 its identity element. The logarithm is a group isomorphism...