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O'Nan Group
In the area of abstract algebra known as group theory, the O'Nan group ''O'N'' or O'Nan–Sims group is a sporadic simple group of order : 460,815,505,920 = 2934573111931 ≈ 5. History ''O'N'' is one of the 26 sporadic groups and was found by in a study of groups with a Sylow 2-subgroup of " Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2''n''Z ×Z/2''n''Z ×Z/2''n''Z).PSL3(F2). The following simple groups have Sylow 2-subgroups of Alperin type: * For the Chevalley group ''G''2(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not split. * For the Steinberg group 3''D''4(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not split. * For the alternating group A8, ''n = 1'' and the extension splits. * For the O'Nan group, ''n'' = 2 and the extension does not split. * For the Higman-Sims group, ''n'' = 2 and the extension splits. The Schur multiplier has order 3, and its outer a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. So in the algebraic structure of groups, H is a subquotient of G if there exists a subgroup G' of G and a normal subgroup G'' of G' so that H is isomorphic to G'/G''. In the literature about sporadic groups wordings like "H is involved in G" can be found with the apparent meaning of "H is a subquotient of G". As in the context of subgroups, in the context of subquotients the term ''trivial'' may be used for the two subquotients G and \ which are present in every group G. A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e. g., Harish-Chandra's subquotient theorem. p. 310 Example There are subquot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry In Mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured object ''X'' of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if ''X'' is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object ''X'' is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry). In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above. Symmetry in geometry The types of symmetry considered in basic geometry include ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monstrous Moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, and the phrase was coined by John Conway and Simon P. Norton in 1979. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with Index of a subgroup, index 2 and has therefore factorial, ''n''!/2 elements. It is the kernel (algebra), kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian group, abelian if and only if and simple group, simple if and only if or . A5 is the smallest non-abelian simple group, having order of a group, order 60, and thus the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M11
In the area of modern algebra known as group theory, the Mathieu group ''M''11 is a sporadic simple group of order : 7,920 = 111098 = 2432511. History and properties ''M''11 is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. ''M''11 is a sharply 4-transitive permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ... on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system. Representations ''M''11 has a sharply 4-transitive permutation r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J1
In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of Order (group theory), order : 175,560 = 233571119 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups. In 1986 Robert Arnott Wilson, Robert A. Wilson showed that ''J1'' cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the pariah group, pariahs. Properties The smallest faithful complex representation of ''J1'' has dimension 56. ''J1'' can be characterized abstractly as the unique simple group with abelian Sylow theorems, 2-Sylow subgroups and with an Involution (mathematics), involution whose centralizer is isomorphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outer Automorphism Group
In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a trivial center, then is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' strictly. In other words, ''H'' is a maximal element of the partially ordered set of subgroups of ''G'' that are not equal to ''G''. Maximal subgroups are of interest because of their direct connection with primitive permutation representations of ''G''. They are also much studied for the purposes of finite group theory: see for example Frattini subgroup, the intersection of the maximal subgroups. In semigroup theory, a maximal subgroup of a semigroup ''S'' is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of ''S'' which is not properly contained in another subgroup of ''S''. Notice that, here, there is no requirement that a maximal subgroup be proper, so if ''S'' is in fact a group then its uni ... [...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] |
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] |
