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Subgroup Method
The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ... of an element. It doesn't always return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this: function operate(element, generator) function subgroup(g) sequence := (set of subgroups that will be used, depending on the method.) word := [] for subgroup in sequence coset_representatives := [] for operation in coset_representatives if operate(g, operation) is in the next subgroup then append operation onto word g = operate(g, operation) bre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
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] |
Word (group Theory)
In group theory, a word is any written product of group elements and their inverses. For example, if ''x'', ''y'' and ''z'' are elements of a group ''G'', then ''xy'', ''z''−1''xzz'' and ''y''−1''zxx''−1''yz''−1 are words in the set . Two different words may evaluate to the same value in ''G'', or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. Definitions Let ''G'' be a group, and let ''S'' be a subset of ''G''. A word in ''S'' is any expression of the form :s_1^ s_2^ \cdots s_n^ where ''s''1,...,''sn'' are elements of ''S'', called generators, and each ''εi'' is ±1. The number ''n'' is known as the length of the word. Each word in ''S'' represents an element of ''G'', namely the product of the expression. By convention, the unique Uniqueness of identity element and inverses identity element can be represented by the empty word, which is the unique word ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup Series
In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the subgroup method. Subgroup series are a special example of the use of filtrations in abstract algebra. Definition Normal series, subnormal series A subnormal series (also normal series, normal tower, subinvariant series, or just series) of a group ''G'' is a sequence of subgroups, each a normal subgroup of the next one. In a standard notation :1 = A_0\triangleleft A_1\triangleleft \cdots \triangleleft A_n = G. There is no requirement made that ''A''''i'' be a normal subgroup of ''G'', only a normal subgroup of ''A''''i'' +1. The quotient groups ''A''''i'' +1/''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
