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Complement (group Theory)
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup ''H'' in a group ''G'' is a subgroup ''K'' of ''G'' such that :G = HK = \ \text H\cap K = \. Equivalently, every element of ''G'' has a unique expression as a product ''hk'' where ''h'' ∈ ''H'' and ''k'' ∈ ''K''. This relation is symmetrical: if ''K'' is a complement of ''H'', then ''H'' is a complement of ''K''. Neither ''H'' nor ''K'' need be a normal subgroup of ''G''. Properties * Complements need not exist, and if they do they need not be unique. That is, ''H'' could have two distinct complements ''K''1 and ''K''2 in ''G''. * If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group. * If ''K'' is a complement of ''H'' in ''G'' then ''K'' forms both a left and right transversal of ''H''. That is, the elements of ''K'' form a complete set of representatives of both the left and right cosets of ''H'' ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zappa–Szép Product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904). Internal Zappa–Szép products Let ''G'' be a group with identity element ''e'', and let ''H'' and ''K'' be subgroups of ''G''. The following statements are equivalent: * ''G'' = ''HK'' and ''H'' ∩ ''K'' = * For each ''g'' in ''G'', there exists a unique ''h'' in ''H'' and a unique ''k'' in ''K'' such that ''g = hk''. If either (and hence both) of these statements hold, then ''G'' is said to be an internal Zappa–Szép product of ''H'' and ''K''. Examples Let ''G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complemented Group
In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways. In , a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following and . The following are equivalent for any finite group ''G'': * ''G'' is complemented * ''G'' is a subgroup of a direct product of groups of square-free order (a special type of Z-group) * ''G'' is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), . Later, in , a group is said to be complemented if the lattice of subgroups is a complemented lattice In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''& ..., that is, if for every subgroup ''H'' there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
