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Power Closed
In 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 ... a p-group G is called power closed if for every Section (group theory), section H of G the product of p^k powers is again a p^kth power. Regular p-groups are an example of power closed groups. On the other hand, powerful p-groups, for which the product of p^k powers is again a p^kth power are not power closed, as this property does not hold for all sections of powerful p-groups. The power closed 2-groups of exponent at least eight are described in . References * Group theory P-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] |
P-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ''n'' such that the product of ''pn'' copies of ''g'', and not fewer, is equal to the identity element. The orders of different elements may be different powers of ''p''. Abelian ''p''-groups are also called ''p''-primary or simply primary. A finite group is a ''p''-group if and only if its order (the number of its elements) is a power of ''p''. Given a finite group ''G'', the Sylow theorems guarantee the existence of a subgroup of ''G'' of order ''pn'' for every prime power ''pn'' that divides the order of ''G''. Every finite ''p''-group is nilpotent. The remainder of this article deals with finite ''p''-groups. For an example of an infinite abelian ''p''-group, see Prüfer group, and for an example of an infinite simple ''p''-grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section (group Theory)
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. In the literature about sporadic groups wordings like «H is involved in G» can be found with the apparent meaning of «H is a subquotient of G». A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem. p. 310 Examples Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 as "pariah groups". Order relation The relation ''subquotient of'' is an order relation. Proof of transitivity for groups Let H'/H'' be subquotient of H, furthermore H := G'/G'' be subquotient of G and \varphi \colon G' \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular P-group
In mathematical finite group theory, the concept of regular ''p''-group captures some of the more important properties of abelian ''p''-groups, but is general enough to include most "small" ''p''-groups. Regular ''p''-groups were introduced by . Definition A finite ''p''-group ''G'' is said to be regular if any of the following equivalent , conditions are satisfied: * For every ''a'', ''b'' in ''G'', there is a ''c'' in the derived subgroup ''H''′ of the subgroup ''H'' of ''G'' generated by ''a'' and ''b'', such that ''a''''p'' · ''b''''p'' = (''ab'')''p'' · ''c''''p''. * For every ''a'', ''b'' in ''G'', there are elements ''c''''i'' in the derived subgroup of the subgroup generated by ''a'' and ''b'', such that ''a''''p'' · ''b''''p'' = (''ab'')''p'' · ''c''1''p'' ⋯ ''c''k''p''. * For every ''a'', ''b'' in ''G'' and every positive integer ''n'', there are elements ''c''''i'' in the derived subgroup of the subgroup generated by ''a'' and ''b'' such that ''a''''q'' · ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powerful P-group
In mathematics, in the field of group theory, especially in the study of ''p''-groups and pro-''p''-groups, the concept of powerful ''p''-groups plays an important role. They were introduced in , where a number of applications are given, including results on Schur multipliers. Powerful ''p''-groups are used in the study of automorphisms of ''p''-groups , the solution of the restricted Burnside problem , the classification of finite ''p''-groups via the coclass conjectures , and provided an excellent method of understanding analytic pro-''p''-groups . Formal definition A finite ''p''-group G is called powerful if the commutator subgroup ,G/math> is contained in the subgroup G^p = \langle g^p , g\in G\rangle for odd p, or if ,G/math> is contained in the subgroup G^4 for p=2. Properties of powerful ''p''-groups Powerful ''p''-groups have many properties similar to abelian groups, and thus provide a good basis for studying ''p''-groups. Every finite ''p''-group can be express ... [...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] |
