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Metacyclic Group
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ''G'' for which there is a short exact sequence :1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\, where ''H'' and ''K'' are cyclic. Equivalently, a metacyclic group is a group ''G'' having a cyclic normal subgroup ''N'', such that the quotient ''G''/''N'' is also cyclic. Properties Metacyclic groups are both supersolvable and metabelian. Examples * Any cyclic group is metacyclic. * The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups. * The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.) * Every finite group of squarefree order is metacyclic. * More generally every Z-group In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidirect Product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. 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 normal subgroup , the following statements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: * in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. * in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. * in the study of ordered groups, a Z-group or \mathbb Z-group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementarily equivalent to the integers (\mathbb Z,+,<). Z-groups are an alternative presentation of . * occasionally, (Z)-group is used to mean a , a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-free Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of 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 or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dicyclic Group
In group theory, a dicyclic group (notation Dic''n'' or Q4''n'', Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST) is a particular kind of non-abelian group of order 4''n'' (''n'' > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2''n'', giving the name ''di-cyclic''. In the notation of exact sequences of groups, this extension can be expressed as: :1 \to C_ \to \mbox_n \to C_2 \to 1. \, More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group. Definition For each integer ''n'' > 1, the dicyclic group Dic''n'' can be defined as the subgroup of the unit quaternions generated by :\begin a & = e^\frac = \cos\frac + i\sin\frac \\ x & = j \end More abstractly, one can define the dicyclic group Dic''n'' as the group with the following presentation :\operatorname_n = \left\langle a, x \mid a^ = 1,\ x^2 = a^n,\ x^ax = a^\right\rangle.\,\! Some things to note which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting oppo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product Of Groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is associative. ;Identity: The direct product has an identity element, namely , where is the identity e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabelian
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group ''G'' is metabelian if and only if there is an abelian normal subgroup ''A'' such that the quotient group ''G/A'' is abelian. Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms. Metabelian groups are solvable. In fact, they are precisely the solvable groups of derived length at most 2. Examples * Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2. * If ''F'' is a field, the group of affine maps x \mapsto ax+b (where ''a'' ≠0) acting on ''F'' is metabelian. Here the abelian normal subgroup is the group of pure translations x\mapsto x+b , and the abelian quotient group is isomorphic to the group of homotheties x\mapsto ax . If ''F'' is a finite field with ''q'' elements, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersolvable Group
In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. Definition Let ''G'' be a group. ''G'' is supersolvable if there exists a normal series :\ = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_ \triangleleft H_s = G such that each quotient group H_/H_i \; is cyclic and each H_i is normal in G. By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a subnormal series with each quotient cyclic, but there is no requirement that each H_i be normal in G. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points, A_4, is solvable but not supersolvable. Basic Properties Some facts about supersolvable groups: * Supersolvabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
