Abelian group information

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

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...

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...

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...

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...

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...

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...

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: the sum h +...

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...

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, 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...

{C} ^{\times }} is abelian, it follows that T {\displaystyle \mathbb {T} } is as well. A unit complex number in the circle group represents a rotation...

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

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...

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...

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

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)...

In group theory, a Dedekind group is a group G such that every subgroup of G is normal. All abelian groups are Dedekind groups. A non-abelian Dedekind...

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...

an example, the direct sum of two abelian groups A {\displaystyle A} and B {\displaystyle B} is another abelian group A ⊕ B {\displaystyle A\oplus B} consisting...