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Absolutely Simple Group
In mathematics, in the field of group theory, a group (mathematics), group is said to be absolutely simple if it has no proper nontrivial serial subgroups.. That is, G is an absolutely simple group if the only serial subgroups of G are \ (the trivial subgroup), and G itself (the whole group). In the finite case, a group is absolutely simple if and only if it is simple group, simple. However, in the infinite case, absolutely simple is a stronger property than simple. The property of being strictly simple is somewhere in between. See also * Ascendant subgroup * Strictly simple group References Properties of groups {{group-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serial Subgroup
In the mathematical field of group theory, a subgroup ''H'' of a given group ''G'' is a serial subgroup of ''G'' if there is a chain ''C'' of subgroups of ''G'' extending from ''H'' to ''G'' such that for consecutive subgroups ''X'' and ''Y'' in ''C'', ''X'' is a normal subgroup of ''Y''. The relation is written ''H ser G'' or ''H is serial in G''. If the chain is finite between ''H'' and ''G'', then ''H'' is a subnormal subgroup of ''G''. Then every subnormal subgroup of ''G'' is serial. If the chain ''C'' is well-ordered and ascending, then ''H'' is an ascendant subgroup of ''G''; if descending, then ''H'' is a descendant subgroup of ''G''. If ''G'' is a locally finite group, then the set of all serial subgroups of ''G'' form a complete sublattice in the lattice of all normal subgroups of ''G''. See also *Characteristic subgroup * Normal closure *Normal core In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The date of incorporation is listed as 1999 by Companies House of Gibraltar, who class it as a holding company; however it is understood that SIMPLE Group's business and trading activities date to the second part of the 90s, probably as an incorporated body. SIMPLE Group Limited is a conglomerate that cultivate secrecy, they are not listed on any Stock Exchange and the group is owned by a complicated series of offshore trust An offshore trust is a conventional trust that is formed under the laws of an offshore jurisdiction. Generally offshore trusts are similar in nature and effect to their onshore counterparts; they involve a settlor transferring (or 'settling') a ...s. The Sunday Times stated that SIMPLE Group's interests could be eval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ascendant Subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal. Here are some properties of ascendant subgroups: * Every subnormal subgroup is ascendant; every ascendant subgroup is serial. * In a finite group, the properties of being ascendant and subnormal are equivalent. * An arbitrary intersection of ascendant subgroups is ascendant. * Given any subgroup, there is a minimal ascendant subgroup containing it. See also * Descendant subgroup In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec ... References * * S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Simple Group
In mathematics, in the field of group theory, a group is said to be strictly simple if it has no proper nontrivial ascendant subgroups. That is, G is a strictly simple group if the only ascendant subgroups of G are \ (the trivial subgroup), and G itself (the whole group). In the finite case, a group is strictly simple if and only if it is simple. However, in the infinite case, strictly simple is a stronger property than simple. See also * Serial subgroup * Absolutely simple group In mathematics, in the field of group theory, a group (mathematics), group is said to be absolutely simple if it has no proper nontrivial serial subgroups.. That is, G is an absolutely simple group if the only serial subgroups of G are \ (the trivia ... ReferencesSimple Group Encyclopedia of Mathematics, retrieved 1 January 2012 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
