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- Abelian group theory

Elementary abelian group

In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime,

Corestriction

In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a

Locally compact abelian group

In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For ex

Cotorsion group

In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is , this says that for all torsion-free groups . It suffices to

Torsion (algebra)

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the

Pure subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and relat

Divisible group

In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every ele

Group of rational points on the unit circle

In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be

Prüfer theorems

In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.

Butler group

In mathematics, a Butler group is a group that is the image of a completely decomposable abelian group of finite rank. They were introduced by M. C. R. Butler.

Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A). An abelian group A is

Torsion abelian group

In abstract algebra, a torsion 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 group.

Minkowski addition

In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set Analogously,

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th r

Locally cyclic group

In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

Ulm's theorem

No description available.

Torsion-free abelian group

In mathematics, specifically in abstract 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 commutati

Algebraically compact group

In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup. Equivalent characte

Essential subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept

Abelian 2-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 by Alexander Grothendieck while studying abstr

Basic subgroup

In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulik

Slender group

In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.

Bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring

Topological abelian group

In mathematics, a 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 are continuous, a

Free abelian group

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 associative, commutative, and invertible. A basi

Herbrand quotient

In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the correspon

Cyclic group

In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible

Height (abelian group)

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solut

Norm (abelian group)

In mathematics, specifically abstract algebra, if is an (abelian) group with identity element then is said to be a norm on if: 1.
* Positive definiteness: , 2.
* Subadditivity: , 3.
* Inversion (Sy

Baer–Specker group

In mathematics, in the field of group theory, the Baer–Specker group, or Specker group, named after Reinhold Baer and Ernst Specker, is an example of an infinite Abelian group which is a building bloc

Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abe

Abelian Lie group

In geometry, an abelian Lie group is a Lie group that is an abelian group. A connected abelian real Lie group is isomorphic to . In particular, a connected abelian (real) compact Lie group is a torus;

Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that

Abelian group

In 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 not depend on the order in which they are w

Primary cyclic group

In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p.That is, it is a cyclic group of order pm, Cpm, for some prime number p, and

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