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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. 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». 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 Examples Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 as "pariah groups". Order relation The relation ''subquotient of'' is an order relation. Proof of transitivity for groups Let H'/H'' be subquotient of H, furthermore H := G'/G'' be subquotient of G and \varphi \colon G' \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monster Group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475971 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the ''happy family'', and the remaining six exceptions ''pariahs''. It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in ''Scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pariah Group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 .... The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family. For example, the orders of ''J''4 and the Lyons Group ''Ly'' are divisible by 37. Since 37 does not divide the order of the monster, these cannot be subquotients of it; thus ''J''4 and ''Ly'' are pariahs. Three other sporadic groups were also shown to be pariahs by Griess in 1982, and the Janko Group J1 was shown to be the final pariah by Robert A. Wilson in 1986. The complete list is shown below. References * * Robert A. Wilson (1986)''Is J1 a subgroup of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcountable
In constructive mathematics, a collection X is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as \exists (I\subseteq).\, \exists f.\, (f\colon I\twoheadrightarrow X), where f\colon I\twoheadrightarrow X denotes that f is a surjective function from a I onto X. The surjection is a member of \rightharpoonup X and here the subclass I of is required to be a set. In other words, all elements of a subcountable collection X are functionally in the image of an indexing set of counting numbers I\subseteq and thus the set X can be understood as being dominated by the countable set . Note that nomenclature of countability and finiteness properties vary substantially, historically. The discussion here concerns the property defined in terms of surjections onto the set in question. Discussion Example An important case is where X denotes some subclass of a bigger class of functions as studied in computability theory. Consider the to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Homomorphism
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of ''choices'' of canonical maps or canonical isomorphisms; for a typical example, see prestack. For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference. Examples *If ''N'' is a normal subgroup of a group ''G'', then there is a canonical surjective group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set. Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets. In the past, "0" was occasionally used as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Relation
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third ossibilityis given". It is a tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
