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Class Of Groups
A class of groups is a set theoretical collection of groups satisfying the property that if ''G'' is in the collection then every group isomorphic to ''G'' is also in the collection. This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativity). Since set theory does not admit the "set of all groups", it is necessary to work with the more general concept of ''class''. Definition A class of groups \mathfrak is a collection of groups such that if G \in \mathfrak and G\cong H then H \in \mathfrak. Groups in the class \mathfrak~ are referred to as \mathfrak-groups. For a set of groups \mathfrak, we denote by (\mathfrak) the smallest class of groups containing \mathfrak. In particular for a group G, (G) denotes its isomorphism class. Examples The most common examples of classes of groups are: * \emptyset: the empty class of groups * \mathfrak: the class of cyclic groups. * \mathfrak~: the class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Groups
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object *Material properties, properties by which the benefits of one material versus another can be assessed *Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-value p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formation (group Theory)
In group theory, a branch of mathematics, a formation is a class of groups closed under taking images and such that if ''G''/''M'' and ''G''/''N'' are in the formation then so is ''G''/''M''∩''N''. introduced formations to unify the theory of Hall subgroups and Carter subgroups of finite solvable groups. Some examples of formations are the formation of ''p''-groups for a prime ''p'', the formation of π-groups for a set of primes π, and the formation of nilpotent groups. Special cases A Melnikov formation is closed under taking quotients, normal subgroups and group extensions. Thus a Melnikov formation ''M'' has the property that for every short exact sequence :1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1\ ''A'' and ''C'' are in ''M'' if and only if ''B'' is in ''M''.Fried & Jarden (2004) p.344 A full formation is a Melnikov formation which is also closed under taking subgroups. An almost full formation is one which is closed under quotients, direct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation for thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1, or e depending on the context. If the group operation is denoted \, \cdot \, then it is defined by e \cdot e = e. The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. Definitions Given any group G, the group consisting of only the identity element is a subgroup of G, and, being the trivial group, is called the of G. The term, when referred to "G has no nontrivial proper subgroups" refers to the only subgroups of G being the trivial group \ and the group G itself. Properties The trivial group is cyc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of A Group
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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