Perfect Group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., more specifically in group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ..., a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... is said to be perfect if it equals its own commutator subgroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numb ... [...More Info...]       [...Related Items...] picture info Mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, �έχνη ''tiké échne', 'art' or 'cra ... and number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...), formulas and related structures (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...), shapes and spaces in which they are contained (geometry Geometry (from the grc, ... [...More Info...]       [...Related Items...] Direct Product Of Groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., specifically in group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ..., the direct product is an operation that takes two groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ... and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product In ... [...More Info...]       [...Related Items...] picture info Universal Central Extension In mathematical group theory, the Schur multiplier or Schur multiplicator is the second group homology, homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \operatorname(G) of a finite group ''G'' is a finite abelian group whose Periodic group, exponent divides the order of ''G''. If a Sylow subgroup, Sylow ''p''-subgroup of ''G'' is cyclic for some ''p'', then the order of \operatorname(G) is not divisible by ''p''. In particular, if all Sylow subgroup, Sylow ''p''-subgroups of ''G'' are cyclic, then \operatorname(G) is trivial. For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion gro ... [...More Info...]       [...Related Items...] Superperfect Group In mathematics, in the realm of group theory, a group (mathematics), group is said to be superperfect when its first two group homology, homology groups are trivial group, trivial: ''H''1(''G'', Z) = ''H''2(''G'', Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology. Definition The first homology group of a group is the commutator subgroup, abelianization of the group itself, since the homology of a group ''G'' is the homology of any Eilenberg–MacLane space of type ''K''(''G'', 1); the fundamental group of a ''K''(''G'', 1) is ''G'', and the first homology of ''K''(''G'', 1) is then abelianization of its fundamental group. Thus, if a group is superperfect, then it is perfect group, perfect. A finite group, finite perfect group is supe ... [...More Info...]       [...Related Items...] Acyclic Group In mathematics, an acyclic space is a nonempty topological space ''X'' in which cycles are always boundaries, in the sense of Homology (mathematics), homology theory. This implies that integral homology groups in all dimensions of ''X'' are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology, :\tilde_i(X)=0, \quad \forall i\ge -1. It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." The condition of acyclicity on a space ''X'' implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of ''X'' to the circle or to the higher s ... [...More Info...]       [...Related Items...] Order (group Theory) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the order of a finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ... is the number of its elements. If a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order o ... [...More Info...]       [...Related Items...] picture info Divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a divisor of an integer n, also called a factor of n, is an integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ... m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-Eur ... [...More Info...]       [...Related Items...] Solvable Group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., more specifically in the field of group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ..., a solvable group or soluble group is a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... that can be constructed from abelian group In mathematics Mathematics (from Greek: ) includes the study of such to ... [...More Info...]       [...Related Items...] picture info Determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the determinant is a scalar value that is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... of the entries of a square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It .... It allows characterizing some properties of the matrix and the linear map In mathematics Mathematics (from Ancient Greek, ... [...More Info...]       [...Related Items...] picture info Complex Number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the form , 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 are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of 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 c ... [...More Info...]       [...Related Items...] picture info Real Number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., a real number is a value of a continuous quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ... that can represent a distance along a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ... (or alternatively, a quantity that can be represented as an infinite decimal expansion). The adjective ''real'' in this co ... [...More Info...]       [...Related Items...] picture info Projective Special Linear Group In mathematics, especially in the group theory, group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced Group action (mathematics), action of the general linear group of a vector space ''V'' on the associated projective space P(''V''). Explicitly, the projective linear group is the quotient group :PGL(''V'') = GL(''V'')/Z(''V'') where GL(''V'') is the general linear group of ''V'' and Z(''V'') is the subgroup of all nonzero scalar transformations of ''V''; these are quotiented out because they act trivial action, trivially on the projective space and they form the Kernel (algebra), kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of a group, center of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: :PSL(''V'') = ... [...More Info...]       [...Related Items...]