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Lattice Theorem
In group theory, the correspondence theorem (also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem ) states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G containing N, onto the set of all subgroups of the quotient group G/N. The structure of the subgroups of G/N is exactly the same as the structure of the subgroups of G containing N, with N collapsed to the identity element. Specifically, if : ''G'' is a group, : N \triangleleft G, a normal subgroup of ''G'', : \mathcal = \, the set of all subgroups ''A'' of ''G'' that contain ''N'', and : \mathcal = \, the set of all subgroups of ''G''/''N'', then there is a bijective map \phi: \mathcal \to \mathcal such that : \phi(A) = A/N for all A \in \mathcal. One further has that if ''A'' and ''B'' are in \mathcal then * A \subseteq B if and only if A/N \subseteq ... [...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] |
