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Prüfer Theorems
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov. Statement Let ''A'' be an abelian group. If ''A'' is finitely generated then by the fundamental theorem of finitely generated abelian groups, ''A'' is decomposable into a direct sum of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic groups in two special cases. The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a countable abelian ''p''-group whose non-trivial elements have finite ''p''-height is isomorphic to a direct sum of cyclic groups. Examples show that the a ... [...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] |
Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory. In 1915 he began his University studies in Mathematics, Physics and Chemistry in Berlin. After that he started his Doctorate degree with Issai Schur as his advisor at Friedrich Wilhelm University, Berlin. In 1921 he obtained his Doctorate degree. His thesis was named ''Unendliche Abelsche Gruppen von Elementen endlicher Ordnung'' (Infinite abelian groups of elements of finite order). This thesis set the road for his contributions on abelian groups. In 1922 he worked with mathematician Paul Koebe in the University of Jena, and in 1923 he obtained tenure and was at this University until 1927. In that year he moved to Münster University where he worked until the end of his life. His final work was about projective geometry, but it was posth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Abelian Group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a ''generating set'' of G or that x_1,\dots, x_s ''generate'' G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group \left(\mathbb,+\right) of rational numbers is not finitely generat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Finitely Generated Abelian Groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a ''generating set'' of G or that x_1,\dots, x_s ''generate'' G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group \left(\mathbb,+\right) of rational numbers is not finitely generat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent dividing its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, see Golod–Shafarevich theorem, and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with finite expo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponent (group Theory)
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent dividing its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, see Golod–Shafarevich theorem, and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with finite exponen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height (abelian Group)
In mathematics, the height of an element ''g'' of an abelian group ''A'' is an invariant that captures its divisibility properties: it is the largest natural number ''N'' such that the equation ''Nx'' = ''g'' has a solution ''x'' ∈ ''A'', or the symbol ∞ if there is no such ''N''. The ''p''-height considers only divisibility properties by the powers of a fixed prime number ''p''. The notion of height admits a refinement so that the ''p''-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants. Definition of height Let ''A'' be an abelian group and ''g'' an element of ''A''. The ''p''-height of ''g'' in ''A'', denoted ''h''''p''(''g''), is the largest natural number ''n'' such that the equation ''p''''n''''x'' = ''g'' has a solution in ''x'' ∈ ''A'', or the symbol ∞ if a solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
