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Abnormal Subgroup
In mathematics, specifically group theory, an abnormal subgroup is a subgroup ''H'' of a group ''G'' such that for all ''x'' in ''G'', ''x'' lies in the subgroup generated by ''H'' and ''H''''x'', where ''H''''x'' denotes the conjugate subgroup In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ... ''xHx''−1. Here are some facts relating abnormality to other subgroup properties: * Every abnormal subgroup is a self-normalizing subgroup, as well as a contranormal subgroup. * The only normal subgroup that is also abnormal is the whole group. * Every abnormal subgroup is a weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup. * Every abnormal subgroup is a pronormal subgroup, and hence a weakly pronormal subgroup, a paranormal subgroup, and 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 ... [...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] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Subgroup
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^. for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-normalizing Subgroup
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutativity, commutes with each element of ''S'', or equivalently, such that Conjugation (group theory), conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the Set (mathematics), 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 (mathematics), ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contranormal Subgroup
In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group. Some facts: * Every subgroup of a finite group is a contranormal subgroup of a subnormal subgroup. In general, every subgroup of a group is a contranormal subgroup of a descendant subgroup In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec .... * Every abnormal subgroup is contranormal. References Bibliography * Subgroup properties {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pronormal Subgroup
In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, . A subgroup is pronormal if each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate. That is, ''H'' is pronormal in ''G'' if for every ''g'' in ''G'', there is some ''k'' in the subgroup generated by ''H'' and ''H''''g'' such that ''H''''k'' = ''H''''g''. (Here ''H''''g'' denotes the conjugate subgroup ''gHg''''-1''.) Here are some relations with other subgroup properties: *Every normal subgroup is pronormal. *Every Sylow subgroup is pronormal. *Every pronormal subnormal subgroup is normal. *Every abnormal subgroup is pronormal. *Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property. *Every pronormal subgroup is paranormal Paranormal events are purpo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paranormal Subgroup
In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it ''within'' that subgroup. In symbols, H is paranormal in G if given any g in G, the subgroup K generated by H and H^g is also equal to H^K. Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups. Here are some facts relating paranormality to other subgroup properties: * Every pronormal subgroup, and hence, every normal subgroup and every abnormal subgroup, is paranormal. * Every paranormal subgroup is a polynormal subgroup In mathematics, in the field of group theory, a subgroup of a group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural i .... * In finite solvable groups, every polynormal subgroup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynormal Subgroup
In mathematics, in the field of group theory, a subgroup of a group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ... is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated. In symbols, a subgroup H of a group G is called polynormal if for any g \in G the subgroup K = H^ is the same as H^. Here are the relationships with other subgroup properties: * Every weakly pronormal subgroup is polynormal. * Every paranormal subgroup is polynormal. Subgroup properties {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
