Abelian group in the context of Algebra over a field

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

The ring of integers of a number field K, denoted by O_K, is the intersection of K and A: it can also be characterized as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer if and only if the ring ${\displaystyle \mathbb {Z} [\alpha ]}$ is finitely generated as an abelian group, which is to say, as a ${\displaystyle \mathbb {Z} }$-module.

In abstract algebra, an abelian group ${\displaystyle (G,+)}$ is called finitely generated if there exist finitely many elements ${\displaystyle x_{1},\dots ,x_{s}}$ in ${\displaystyle G}$ such that every ${\displaystyle x}$ in ${\displaystyle G}$ can be written in the form ${\displaystyle x=n_{1}x_{1}+n_{2}x_{2}+\cdots +n_{s}x_{s}}$ for some integers ${\displaystyle n_{1},\dots ,n_{s}}$. In this case, we say that the set ${\displaystyle \{x_{1},\dots ,x_{s}\}}$ is a generating set of ${\displaystyle G}$ or that ${\displaystyle x_{1},\dots ,x_{s}}$ generate ${\displaystyle G}$. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.

Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication.

Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets A and B of elements from an abelian group G,

and the h-fold sumset of A,

In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup ${\displaystyle H\leq G}$ such that H has property P.

Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

More formally, a ring is a set that is endowed with two binary operations (addition and multiplication) such that the ring is an abelian group with respect to addition. The multiplication is associative, is distributive over the addition operation, and has a multiplicative identity element. Some authors apply the term ring to a further generalization, often called a rng, that omits the requirement for a multiplicative identity, and instead call the structure defined above a ring with identity.

An additive group is a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.

This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the additive group of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible elements.

In mathematics, especially group theory, two elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group are conjugate if there is an element ${\displaystyle g}$ in the group such that ${\displaystyle b=gag^{-1}.}$ This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under ${\displaystyle b=gag^{-1}}$ for all elements ${\displaystyle g}$ in the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

In mathematics, the sciences, and engineering, Fourier analysis (/ˈfʊrieɪ, -iər/) is the study of the way general functions on the real line, circle, integers, finite cyclic group or general locally compact Abelian group may be represented or approximated by sums of trigonometric functions or more conveniently, complex exponentials. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Fourier analysis has applications in many areas of pure and applied mathematics, in the sciences and in engineering. The process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One can then re-synthesize the same sound by mixing purely harmonic sounds with frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.

In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently ${\displaystyle \mathbb {Z} }$_n or Z_n, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is 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), and every finitely generated abelian group is a direct product of cyclic groups.

In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour.

The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.

In 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 least one pair of elements a and b of G, such that 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 and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order).

In mathematics, the sumset of two subsets A and B of an (additive) abelian group is formed by adding each element of A to each element of B:$${\displaystyle A+B=\{a+b\mid a\in A,\ b\in B\}.}$$

In geometry, the Minkowski sum of two subsets A and B of a Euclidean space is the set of the points whose position vectors form the sumset of the position vectors of A and B. The Minkowski sum depends on the choice of an origin in the Euclidean space. As a change of origin amounts to translate the Minkowski sum, the Minkowski sum is defined up to a translation, and its shape and orientation are well defined.

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.

Intuitively, a 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 elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

In mathematics, more 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, a solvable group is a group whose derived series terminates in the trivial subgroup.

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 the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive-or operations on two-bit binary values, or more abstractly as ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] , meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter ${\displaystyle V}$ or as ${\displaystyle K_{4}}$.

The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.

Named after the 19th-century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group – such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center – can be discovered from its Cayley table.

A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication: