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Conjugation Map
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group. Definition If is a group and is an element of (alternatively, if is a ring, and is a unit), then the function :\begin \varphi_g\colon G&\to G \\ \varphi_g(x)&:= g^xg \end is called (right) conjugation by (see also conjugacy class). This function is an endomorphism of : for all x_1,x_2\in G, :\varphi_g(x_1 x_2) = g^ x_1 x_2g = \left(g^ x_1 g\right)\left(g^ x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2), where the second equality is given by the insertion of the identity between x_1 and x_2. Furthermore, it has a left and ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Mapping
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we have (whe ... [...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] |
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 symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frattini Subgroup
In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is defined by \Phi(G)=G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after Giovanni Frattini, who defined the concept in a paper published in 1885. Some facts * \Phi(G) is equal to the set of all non-generators or non-generating elements of . A non-generating element of is an element that can always be removed from a generating set; that is, an element ''a'' of such that whenever is a generating set of containing ''a'', X \setminus \ is also a generating set of . * \Phi(G) is always a characteristic subgroup of ; in particular, it is always a normal subgroup of . * If is finite, then \Phi(G) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centralizer And Normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', or equivalently, such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and normaliz ... [...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] |
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] |
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] |
Isomorphism Theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. History The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential ''Moderne Algebra'' the first abstract algebra textbook that took the groups-rings-fields a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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