Simple Group 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 simple group is a nontrivial 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 ... whose only normal subgroup 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), ...s are the trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathem ... [...More Info...]       [...Related Items...] picture info 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 (calculus and mathematical analysis, analysis). There is no general consensus about its exact scope or epistemology, epistemological status. Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of mathematical object, abstract objects. These objects are either abstractions from nature (such as natural numbers or "a line (mathematics), line"), or (in modern mathematics) abstract entities that are defined by their basic properties, called axioms. A proof (mathematics), proof consists of a succession of applications of some inference rule, deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are conside ... [...More Info...]       [...Related Items...] picture info Alternating Group 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 ..., an alternating group is the 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 ... of even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ord ...s of a finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structur ... [...More Info...]       [...Related Items...] Quasithin Group 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 quasithin group is a finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ... simple group In mathematics, a simple group is a nontrivial Group (mathematics), group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgrou ... that resembles a group of Lie type In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical str ... [...More Info...]       [...Related Items...] Daniel Gorenstein Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician A mathematician is someone who uses an extensive knowledge of 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 ( .... He earned his undergraduate and graduate degrees at Harvard University Harvard University is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly tw ..., where he earned his Ph.D. in 1950 under Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire The Russian Empire, . was a historical empire that extended across Eurasia Eurasia () is the largest continental area on Earth, comprising all of Europe and Asia. Pri ..., introducing in hi ... [...More Info...]       [...Related Items...] Isomorphism 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 ..., an isomorphism is a structure-preserving mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ... between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: wikt:ἴσος, ἴσος ''isos'' "equal", and wikt:μορφή, μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that ... [...More Info...]       [...Related Items...] picture info Permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , 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 s, which are selections of some members of a set regardless of order. For example, written as s, 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. s 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 s is an important topic in the fields of and . Permutations are used in almost every branch of mathematics, and in many other fields of science. In , they are used for analyzing s; in , for describing states of particles; and in , fo ... [...More Info...]       [...Related Items...] picture info Up To Two mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ... objects ''a'' and ''b'' are called equal up to an equivalence relation 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). ... ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes 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). I ... of ''a'' and ''b'' with respec ... [...More Info...]       [...Related Items...] Composition Series 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), rings, field (mathema ..., a composition series provides a way to break up an algebraic structure 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 ..., such as 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 ... or a module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and en ... [...More Info...]       [...Related Items...] List Of Finite Simple Groups In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order (group theory), order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small group representation, representations, and lists of all duplicates. Summary The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = ''A''3(2) and ''A''2(4) both have order 20160, and that the group ''Bn''(''q'' ... [...More Info...]       [...Related Items...] Torsion (algebra) 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 ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ..., a torsion element is an element of a module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ... that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule In mathematics ... [...More Info...]       [...Related Items...] picture info Thompson 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 ..., the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three 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 ..., commonly denoted F \subseteq T \subseteq V, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, ''F'' is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and ''F'' in particular, have a ... [...More Info...]       [...Related Items...] Higman 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 ..., the Higman group, introduced by , was the first example of an infinite finitely presented group In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ... with no non-trivial finite quotients. The quotient by the maximal proper normal subgroup 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 a finitely generated infinite ... [...More Info...]       [...Related Items...]