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Derived Group
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group. The commutator subgroup is important because it is the Universal property, smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian group, abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is [g,h] = g^h^gh. The commutator [g,h] is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg[g,h]. However, the notation is somewhat arbitrary and there is a n ... [...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] |
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflective Subcategory
In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A'' is said to be coreflective in ''B'' when the inclusion functor has a right adjoint. Informally, a reflector acts as a kind of completion operation. It adds in any "missing" pieces of the structure in such a way that reflecting it again has no further effect. Definition A full subcategory A of a category B is said to be reflective in B if for each B- object ''B'' there exists an A-object A_B and a B- morphism r_B \colon B \to A_B such that for each B-morphism f\colon B\to A to an A-object A there exists a unique A-morphism \overline f \colon A_B \to A with \overline f\circ r_B=f. : The pair (A_B,r_B) is called the A-reflection of ''B''. The morphism r_B is called the A-reflection arrow. (Although often, for the sake of brevity, we spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Abelian Groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is the trivial group which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a full subcategory of Grp, the category of ''all'' groups. The main difference between Ab and Grp is that the sum of two homomorphisms ''f'' and ''g'' between abelian groups is again a group homomorphism: :(''f''+''g'')(''x''+''y'') = ''f''(''x''+''y'') + ''g''(''x''+''y'') = ''f''(''x'') + ''f''(''y'') + ''g''(''x'') + ''g''(''y'') : = ''f''(''x'') + ''g''(''x'') + ''f''(''y'') + ''g''(''y'') = (''f''+''g'')(''x'') + (''f''+''g'')(''y'') The third e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Core
In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite derived series stabilizes for any group. A group whose perfect core is trivial is termed a hypoabelian group. Every solvable group is hypoabelian, and so is every free group. More generally, every residually solvable group is hypoabelian. The quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ... of a group ''G'' by its perfect core is hypoabelian, and is called the hypoabelianization of ''G''. References Functional subgroups Group theory Solvable groups { ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Recursion
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then is also true. Then transfinite induction tells us that is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that is true. * Successor case: Prove that for any , follows from (and, if necessary, for all ). * Limit case: Prove that for any |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup equals the group), or equivalently one such that ''G''ab = (its abelianization is trivial). Examples The smallest (non-trivial) perfect group is the alternating group ''A''5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Conversely, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing \left(\begin-1 & 0 \\ 0 & -1\end\right) = \left(\begin4 & 0 \\ 0 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Central Series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algrebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable. Definition A central series is a sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal 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] |
Fully Characteristic Subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. Definition A subgroup of a group is called a characteristic subgroup if for every automorphism of , one has ; then write . It would be equivalent to require the stronger condition = for every automorphism of , because implies the reverse inclusion . Basic properties Given , every automorphism of induces an automorphism of the quotient group , which yields a homomorphism . If has a unique subgroup of a given index, then is characteristic in . Related concepts Normal subgroup A subgroup of that is invariant under all inner automorphisms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
