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Quasidihedral Group
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer ''n'' greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2''n'' which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text ''Endliche Gruppen'', this group is called a "Quasidiedergruppe". In Daniel Gorenstein's text, ''Finite Groups'', this group is called the "semidihedral group". Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same presentation for this group: :\langle r,s \mid r^ = s^2 = 1,\ srs = r^\rangle\,\!. The other non-abelian 2-group with cyclic subgroup of index 2 is not gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cayley Graph Of The Quasidihedral Group Of Order 16
Cayley may refer to: __NOTOC__ People * Cayley (surname) * Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow * Cayley Mercer (born 1994), Canadian women's ice hockey player Places * Cayley, Alberta, Canada, a hamlet ** Cayley/A. J. Flying Ranch Airport * Mount Cayley, a volcano in southwestern British Columbia, Canada * Cayley Glacier, Graham Land, Antarctica * Cayley (crater), a lunar crater Other uses * Cayley baronets, a title in the Baronetage of England * Cayley computer algebra system, designed to solve mathematical problems, particularly in group theory See also * W. Cayley Hamilton (died 1891), Canadian barrister and politician * Caylee (name), given name * Cèilidh, traditional Scottish or Irish social gathering * Kaylee, given name * Kaley (other) * Kayleigh (other) "Kayleigh" is a song by the British neo-progressive rock band Marillion. Kayleigh may also refer to: People *Kaylee, a given name with many variants including "Kayleigh", and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bertram Huppert
Bertram Huppert (22 October 1927 – 1 October 2023) was a German mathematician specialised in group theory and the representation theory of finite groups. His ''Endliche Gruppen'' (finite groups) is an influential textbook in group theory, and he has over 50 doctoral descendants. Life Early life and education Bertram Huppert was born in Worms, Germany on 22 October 1927. He went to school in Bonn from 1934 until 1945. In 1950, he wrote his diploma thesis in mathematics at the University of Mainz. The thesis discussed "''nicht fortsetzbare Potenzreihen''" (discontinuous power series), and was written under the direction of Helmut Wielandt. When Wielandt moved to the University of Tübingen in April 1951, Huppert followed him later in the year, and gained his doctorate (as Wielandt's first doctoral student) with the work "''Produkte von paarweise vertauschbaren zyklischen Gruppen''" (products of pairwise permutable cyclic groups), in which he showed, among other things, that s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Simple Group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The date of incorporation is listed as 1999 by Companies House of Gibraltar, who class it as a holding company A holding company is a company whose primary business is holding a controlling interest in the Security (finance), securities of other companies. A holding company usually does not produce goods or services itself. Its purpose is to own Share ...; however it is understood that SIMPLE Group's business and trading activities date to the second part of the 90s, probably as an incorporated body. SIMPLE Group Limited is a conglomerate that cultivate secrecy, they are not listed on any Stock Exchange and the group is owned by a complicated series of offshore trusts. The Sunday Times stated that SIMPLE Group's interests could be eva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Alperin–Brauer–Gorenstein Theorem
In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathedA 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic groups of the same order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points. Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ... of odd order, depending on a certain congruence, or to the Mathieu group M_. proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in , and presented in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Derived Subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ''n'' such that the product of ''pn'' copies of ''g'', and not fewer, is equal to the identity element. The orders of different elements may be different powers of ''p''. Abelian ''p''-groups are also called ''p''-primary or simply primary. A finite group is a ''p''-group if and only if its order (the number of its elements) is a power of ''p''. Given a finite group ''G'', the Sylow theorems guarantee the existence of a subgroup of ''G'' of order ''pn'' for every prime power ''pn'' that divides the order of ''G''. Every finite ''p''-group is nilpotent. The remainder of this article deals with finite ''p''-groups. For an example of an infinite abelian ''p''-group, see Prüfer group, and for an example of an infinite simple ''p' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Group Of Units
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group with identity element , a subgroup , and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Group Presentation
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
