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In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.
Equivalently, a group G is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say G0, ..., Gn such that
A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group.
Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices. [1] The holomorph of a polycyclic group is also such a group of integer matrices. [2]
A polycyclic group G is said to be strongly polycyclic if each quotient Gi+1 / Gi is infinite. Any subgroup of a strongly polycyclic group is strongly polycyclic.
A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite.
In the textbook ( Scott 1964 , Ch 7.1) and some papers, an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite cyclic group.
These groups are particularly interesting because they are the only known examples of Noetherian group rings ( Ivanov 1989 ), or group rings of finite injective dimension.[ citation needed ]
The Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series.
If G is a polycyclic-by-finite group, then the Hirsch length of G is the Hirsch length of a polycyclic normal subgroup H of G, where H has finite index in G. This is independent of choice of subgroup, as all such subgroups will have the same Hirsch length.
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
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.
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination of finitely many elements of the subset and their inverses.
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions.
A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroupF of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroupF*, which is generated by the Fitting subgroup and the components of G.
In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups:
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. 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 mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group.
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots.
In mathematics, a group is supersolvable if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.
In abstract algebra, a chief series is a maximal normal series for a group.
In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem.
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, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal.
In abstract algebra, a uniserial moduleM is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.