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Fully Normalized Subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjective. In symbols, a subgroup H is fully normalized in G if, given an automorphism \sigma of H, there is a g \in G such that the map x \mapsto gxg^, when restricted to H is equal to \sigma. Some facts: * Every group can be embedded as a normal and fully normalized subgroup of a bigger group. A natural construction for this is the holomorph, which is its semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ... with its automorphism group. * A complete group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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 cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Automorphism
In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ... given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the Quotient_group, 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 (ring theory), 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 func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorph (mathematics)
In mathematics, especially in the area of algebra known as group theory, the holomorph of a group G, denoted \operatorname(G), is a group that simultaneously contains (copies of) G and its automorphism group \operatorname(G). It provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. The holomorph can be described as a semidirect product or as a permutation group. Hol(''G'') as a semidirect product If \operatorname(G) is the automorphism group of G then :\operatorname(G)=G\rtimes \operatorname(G) where the multiplication is given by Typically, a semidirect product is given in the form G\rtimes_A where G and A are groups and \phi:A\rightarrow \operatorname(G) is a homomorphism and where the multiplication of elements in the semidirect product is given as :(g,a)(h,b)=(g\phi(a)(h),ab) which is well defined, since \phi(a)\in \operatorname(G) and therefore \phi(a)(h)\in G. For the holomorph, A=\operatorname(G) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Group
In mathematics, a group is said to be complete if every automorphism of is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center. Equivalently, a group is complete if the conjugation map, (sending an element to conjugation by ), is an isomorphism: injectivity implies that only conjugation by the identity element is the identity automorphism, meaning the group is centerless, while surjectivity implies it has no outer automorphisms. Examples As an example, all the symmetric groups, , are complete except when . For the case , the group has a non-trivial center, while for the case , there is an outer automorphism. The automorphism group of a simple group is an almost simple group; for a non- abelian simple group , the automorphism group of is complete. Properties A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the converse need not hold: for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Extension Property
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In an algebraic structure such as a group, a ring, or vector space, an ''automorphism'' is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity morphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
