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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 group (mathematics), groups: * in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic group, 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 linearly ordered group, 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 elementary equivalence, elementarily equivalent to the integers (\mathbb Z,+,<). Z-groups are an alternative presentation of Presburger arithmetic. * occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. Groups whose Sylow subgroups are cyclic :''Usage: , , , , '' In the study of finite groups, a Z-group is a finite group whose Sylow subgr ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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: * Supersol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Frobenius Group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. Structure Suppose ''G'' is a Frobenius group consisting of permutations of a set ''X''. A subgroup ''H'' of ''G'' fixing a point of ''X'' is called a Frobenius complement. The identity element together with all elements not in any conjugate of ''H'' form a normal subgroup called the Frobenius kernel ''K''. (This is a theorem due to ; there is still no proof of this theorem that does not use character theory, although see .) The Frobenius group ''G'' is the semidirect product of ''K'' and ''H'': :G=K\rtimes H. Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel ''K'' is a nilpotent group. If ''H'' has even order then ''K'' is abelian. The Frobenius complement ''H'' has the property th ... [...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] |
Doubly Transitive Permutation Group
A group G acts 2-transitively on a set S if it acts transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq y and w\neq z, there exists a g\in G such that g(x,y) = (w,z). The group action is sharply 2-transitive if such g\in G is unique. A 2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply 2-transitive group. Equivalently, gx = w and gy = z, since the induced action on the distinct set of pairs is g(x,y) = (gx,gy). The definition works in general with ''k'' replacing 2. Such multiply transitive permutation groups can be defined for any natural number ''k''. Specifically, a permutation group ''G'' acting on ''n'' points is ''k''-transitive if, given two sets of points ''a''1, ... ''a''''k'' and ''b''1, ... ''b''''k'' with the property that all the ''a''''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hypocentral Group
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable. Definition A central series is a sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |