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Product Of Group Subsets
In mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group (mathematics), group ''G'', then their product is the subset of ''G'' defined by :ST = \. The subsets ''S'' and ''T'' need not be subgroups for this product to be well defined. The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of ''G''. A lot more can be said in the case where ''S'' and ''T'' are subgroups. The product of two subgroups ''S'' and ''T'' of a group ''G'' is itself a subgroup of ''G'' if and only if ''ST'' = ''TS''. Product of subgroups If ''S'' and ''T'' are subgroups of ''G'', their product need not be a subgroup (for example, two distinct subgroups of order 2 in the symmetric group on 3 symbols). This product is sometimes called the ''Frobenius product''. In general, the product of two subgroups ''S'' and ''T'' is a subgroup if and only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Group
__NOTOC__ In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is permutable in ''G'' . proved that a ''p''-group ''G'' is an Iwasawa group if and only if one of the following cases happens: * ''G'' is a Dedekind group, or * ''G'' contains an abelian normal subgroup ''N'' such that the quotient group ''G/N'' is a cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ... and if ''q'' denotes a generator of ''G/N'', then for all ''n'' ∈ ''N'', ''q''−1''nq'' = ''n''1+''p''''s'' where ''s'' ≥ 1 in general, but ''s'' ≥ 2 for ''p''=2. In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Nap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centralizer And Normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of elements g\in G such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylow P-subgroup
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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frattini's Argument
In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.M. Brescia, F. de Giovanni, M. Trombetti"The True Story Behind Frattini’s Argument" '' Advances in Group Theory and Applications'' 3doi:10.4399/97888255036928/ref> Frattini's argument Statement If G is a finite group with normal subgroup H, and if P is a Sylow ''p''-subgroup of H, then : G = N_G(P)H, where N_G(P) denotes the normalizer of P in G, and N_G(P)H means the product of group subsets. Proof The group P is a Sylow p-subgroup of H, so every Sylow p-subgroup of H is an H-conjugate of P, that is, it is of the form h^Ph for some h \in H (see Sylow theorems). Let g be any element of G. Since H is normal in G, the subgroup g^Pg i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram Rule
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: ''AB'', ''BC'', ''CD'', ''DA''. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, ''AB'' = ''CD'' and ''BC'' = ''DA'', the law can be stated as 2AB^2 + 2BC^2 = AC^2 + BD^2\, If the parallelogram is a rectangle, the two diagonals are of equal lengths ''AC'' = ''BD'', so 2AB^2 + 2BC^2 = 2AC^2 and the statement reduces to the Pythagorean theorem. For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states AB^2 + BC^2 + CD^2+DA^2 = AC^2+BD^2 + 4x^2, where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Moritz Cohn
Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.Independent Early life Cohn was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia (''née'' Cohen), a schoolteacher.Autobiography Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I; he was wounded several times and awarded the Iron Cross. A street in Hamburg is named in memory of his mother.De Morgan When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter. He attended a kindergarten then, in April 1930, m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Isaacs
Irving Martin Isaacs (April 14, 1940 – February 17, 2025) was an American group theorist and representation theorist. He was a professor of mathematics at the University of Wisconsin–Madison until his retirement. Biography Isaacs was born in the Bronx, in New York City, on April 14, 1940. He received a BS from the Polytechnic Institute of Brooklyn in 1960. Isaacs went on to Harvard University for graduate study. He received a masters degree in 1961, and completed his PhD in 1964. His thesis was advised by Richard Brauer, and was titled ''Finite p-solvable linear groups''. After a few years at the University of Chicago as an instructor and visiting assistant professor, Isaacs moved to the University of Wisconsin–Madison in 1969. He was hired as an associate professor, and promoted to full professor in 1971. According to the Mathematics Genealogy Project, he supervised 29 doctoral students over his career. In 2011, Isaacs retired and became a professor emer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Isomorphism Theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. History The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential ''Moderne Algebra'', the first abstract algebra textbook that took the groups- rings-fields approa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theory), intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to family of sets, families of sets and to indexed family, indexed families of sets. By definition, a collection of sets is called a ''family of sets'' (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An \left(A_i\right)_, is by definition a set-valued Function (mathematics), function (that is, it is a function that assigns a set A_i to every ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abuse Of Terminology
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory. ''Abuse of notation'' should be contrasted with ''misuse'' of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration). A related concept is abuse of language or abuse of termin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (group Theory)
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup ''H'' in a group ''G'' is a subgroup ''K'' of ''G'' such that :G = HK = \ \text H\cap K = \. Equivalently, every element of ''G'' has a unique expression as a product ''hk'' where ''h'' ∈ ''H'' and ''k'' ∈ ''K''. This relation is symmetrical: if ''K'' is a complement of ''H'', then ''H'' is a complement of ''K''. Neither ''H'' nor ''K'' need be a normal subgroup of ''G''. Properties * Complements need not exist, and if they do they need not be unique. That is, ''H'' could have two distinct complements ''K''1 and ''K''2 in ''G''. * If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group. * If ''K'' is a complement of ''H'' in ''G'' then ''K'' forms both a left and right transversal of ''H''. That is, the elements of ''K'' form a complete set of representatives of both the left and right cosets of ''H''. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
