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Cayley's Theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose elements are the permutations of the underlying set of . Explicitly, * for each g \in G, the left-multiplication-by- map \ell_g \colon G \to G sending each element to is a permutation of , and * the map G \to \operatorname(G) sending each element to \ell_g is an injective homomorphism, so it defines an isomorphism from onto a subgroup of \operatorname(G). The homomorphism G \to \operatorname(G) can also be understood as arising from the left translation action of on the underlying set . When is finite, \operatorname(G) is finite too. The proof of Cayley's theorem in this case shows that if is a finite group of order , then is isomorphic to a subgroup of the standard symmetric group S_n. But might also be isomorphic to a subgroup o ... [...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] |
Group Homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) where the group operation on the left side of the equation is that of ''G'' and on the right side that of ''H''. From this property, one can deduce that ''h'' maps the identity element ''eG'' of ''G'' to the identity element ''eH'' of ''H'', : h(e_G) = e_H and it also maps inverses to inverses in the sense that : h\left(u^\right) = h(u)^. \, Hence one can say that ''h'' "is compatible with the group structure". Older notations for the homomorphism ''h''(''x'') may be ''x''''h'' or ''x''''h'', though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that ''h''(''x'') becomes simply xh. In areas of mathematics where one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the locally small category \mathcal , one should study the category of all functors of \mathcal into \mathbf (the category of sets with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frucht's Theorem
Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group ''G'' there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to ''G''. Proof idea The main idea of the proof is to observe that the Cayley graph of ''G'', with the addition of colors and orientations on its edges to distinguish the generators of ''G'' from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have ''G'' as its symmetry group., discussion following T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Representation Theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (other).'' In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called ''fields of sets'', closely related to the ''rings of sets'' used by Birkho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wagner–Preston Theorem
In group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x f'' rather than ''f(x)'', and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation ''α'' of a set ''X'' is a function from ''A'' to ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core (group)
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a group ''G'', the normal core or normal interiorRobinson (1996) p.16 of a subgroup ''H'' is the largest normal subgroup of ''G'' that is contained in ''H'' (or equivalently, the intersection of the conjugates of ''H''). More generally, the core of ''H'' with respect to a subset ''S'' ⊆ ''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e. :\mathrm_S(H) := \bigcap_. Under this more general definition, the normal core is the core with respect to ''S'' = ''G''. The normal core of any normal subgroup is the subgroup itself. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group Of Order 6
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abelian group.. For the identification of D3 with S3, and the observation that this group is the smallest possible non-abelian group, sep. 49 This page illustrates many group concepts using this group as example. Symmetry groups The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of highe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise operation, bitwise exclusive or operations on two-bit binary values, or more abstract algebra, abstractly as , the Direct product of groups, direct product of two copies of the cyclic group of Order (group theory), order 2. It was named ''Vierergruppe'' (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4. The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derangements
In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of a set of size ''n'' is known as the subfactorial of ''n'' or the ''n-''th derangement number or ''n-''th de Montmort number. Notations for subfactorials in common use include !''n,'' ''Dn'', ''dn'', or ''n''¡. For ''n'' > 0, the subfactorial !''n'' equals the nearest integer to ''n''!/''e,'' where ''n''! denotes the factorial of ''n'' and ''e'' is Euler's number. The problem of counting derangements was first considered by Pierre Raymond de Montmort in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time. Example Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other's tests. Of course, no student should grade their own test. How many ways could the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
