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Burnside's Theorem
In mathematics, Burnside's theorem in group theory states that if ''G'' is a finite group of order p^a q^b where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers, then ''G'' is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. History The theorem was proved by using the representation theory of finite groups. Several special cases of the theorem had previously been proved by Burnside in 1897, Jordan in 1898, and Frobenius in 1902. John G. Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work in the 1960s and 1970s on the N-group theorem, and this was done explicitly by for groups of odd order, and by for groups of even order. simplified the proofs. Proof The following proof — using more background than Burnside's — is by contradiction. Let ''paqb'' be the smallest product of two prime powers, such that there is a non-solvable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside 2
Burnside may refer to: Places Antarctica * Burnside Ridges, Oates Land Australia * City of Burnside, a local government area of Adelaide, South Australia ** Burnside, South Australia, a suburb of the City of Burnside * Burnside, New South Wales, in the Oatlands suburb of Sydney * Burnside, Queensland, a suburb in the Sunshine Coast Region * Burnside, Victoria, a suburb of Melbourne * Burnside, Western Australia, in the South West region * Lake Burnside, in the Gibson Desert, Western Australia Canada * Burnside, Nova Scotia, an urban neighbourhood in Dartmouth, Nova Scotia * Burnside Drive, a road in Dartmouth, Nova Scotia * Burnside, Colchester County, an unincorporated rural community in Nova Scotia * Burnside Hall, a building on the downtown campus of McGill University, Montreal, Quebec * Burnside, Newfoundland and Labrador, a coastal community in Newfoundland * Burnside, Ontario, in the township of Severn * Burnside River, Nunavut New Zealand * Burnside, Christchur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (group Theory)
In abstract algebra, the center of a group (mathematics), group is the set (mathematics), set of elements that commutative, commute with every element of . It is denoted , from German ''wikt:Zentrum, Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, Z(G)\triangleleft G, and also a characteristic subgroup, characteristic subgroup, but is not necessarily fully characteristic subgroup, fully characteristic. The quotient group, , is group isomorphism, isomorphic to the inner automorphism group, . A group is abelian if and only if . At the other extreme, a group is said to be centerless if is trivial group, trivial; i.e., consists only of the identity element. The elements of the center are central elements. As a subgroup The center of ''G'' is always a subgroup (mathematics), subgroup of . In particular: # contains the identity element of , because it commutes with every element of , by definition: , where is the identity; # If an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Orthogonality Relations
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3). Finite groups Intrinsic statement The space of complex-valued class functions of a finite group ''G'' has a natural inner product: :\left\langle \alpha, \beta \right\rangle := \frac\sum_ \alpha(g) \overline where \overline denotes the complex conjugate of the value of \beta on ''g''. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table: :\left\langle \chi_i, \chi_j \right\rangle = \begin 0& \mbox i \ne j, \\ 1& \mbox i=j. \end For g, h \in G, applying the same inner product to the columns of the character table yields: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations. Applications Characters of irreducible representations encode many important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Of A Subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left Coset, cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or [G:H] or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same cardinality, size as ''H'', the index is related to the order (group theory), orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the Parity (mathematics), even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stabilizer Subgroup
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn i ... [...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] |
Sylow Theorems
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a ''p''-group is a group whose cardinality is a power of p; or equivalently, the order of each group element is some power of p. A Sylow ''p''-subgroup (sometimes ''p''-Sylow subgroup) of a finite group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a ''p''-group and is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written \text_p(G). The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugacy Class
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 \op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent Group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
