|
Modular Subgroup
In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups. By the modular property of groups, every quasinormal subgroup (that is, a subgroup that permutes with all subgroups) is modular. In particular, every 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 i ... is modular. References *. {{DEFAULTSORT:Modular Subgroup Subgroup properties ... [...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] |
Modular Lattice
In the branch of mathematics called order theory, a modular lattice is a lattice (order), lattice that satisfies the following self-duality (order theory), dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety (universal algebra), variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the Second isomorphism theorem, 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Of Subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection. Example The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of order-two elements. The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" ''N''5 as a sublattice. Properties For any ''A'', ''B'', and ''C'' subgroups of a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if ''S'' is a subset of a group ''G'', then , the ''subgroup generated by S'', is the smallest subgroup of ''G'' containing every element of ''S'', which is equal to the intersection over all subgroups containing the elements of ''S''; equivalently, is the subgroup of all elements of ''G'' that can be expressed as the finite product of elements in ''S'' and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If ''G'' = , then we say that ''S'' ''generates'' ''G'', and the elements in ''S'' are called ''generators'' or ''group generators''. If ''S'' is the empty set, then is the trivial group , since we consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasinormal Subgroup
__NOTOC__ In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal subgroup'' was introduced by Øystein Ore in 1937. Two subgroups are said to permute (or commute) if any element from the first subgroup, times an element of the second subgroup, can be written as an element of the second subgroup, times an element of the first subgroup. That is, H and K as subgroups of G are said to commute if ''HK'' = ''KH'', that is, any element of the form hk with h \in H and k \in K can be written in the form k'h' where k' \in K and h' \in H. Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group. The converse is not true. For instance, any extension of a cyclic p-group by another cyclic p-group for the same (odd) prime has the property that all its subgroups are qua ... [...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 for this re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Lattice Of Subgroups")
