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In the area of abstract algebra known as group theory, polycyclic groups form an important subclass of the class of infinite solvable groups. Their study was initiated in the 1930s by Kurt Hirsch as part of a program to investigate whether interesting developments in the theory of finite solvable groups could be carried over, in some form, to classes of infinite solvable groups.

A polycyclic group is defined to be a group $$G$$ that has a chain of subgroups:

for some integer $$n \ge 0,$$ extending from the trivial subgroup $$\{1\}$$ to the whole group $$G$$, such that each is a normal subgroup of its successor (indicated by the symbol $$\triangleleft$$ between the successive terms) and the quotient groups $$G_i/G_{i - 1}$$ are all cyclic groups, for $$i=1,2,\dots,n$$. Note that it is not assumed that the subgroups $$G_i$$ are normal subgroups of the whole group $$G$$, just that each one is a normal subgroup of the next one.

This definition is similar to one of the equivalent conditions commonly used to define a solvable group. According to that definition, a group is solvable if and only if it has a chain of subgroups like ($$), but subject only to the restriction that the quotient groups $$G_i/G_{i - 1}$$ should be abelian rather than cyclic. Since cyclic groups are abelian, every polycyclic group satisfies this condition -- consequently, all polycyclic groups are solvable. However, cyclic groups form a very special class of abelian groups and while all finite abelian groups are direct products of cyclic groups, this is very far from being true for infinite abelian groups. Furthermore, it is known [ref] (and fairly easy to verify) that, for finite groups, the converse to this is also true: that is, every finite solvable group is polycyclic. This means that the term polycyclic is superfluous when one is studying only finite groups, because a finite group is polycyclic if and only if it is solvable.

However, it is not hard to see that a converse like this does not hold for infinite solvable groups. Indeed, many infinite abelian groups are not polycyclic -- the group of rational numbers under addition is a fairly simple example. In fact, it is largely because infinite abelian groups can be so much more complicated than finite abelian groups that it is so difficult to develop a general theory of solvable groups.

Properties of groups defined through chains of subgroups like ($$) and the associated quotient groups were, in essence, first brought to public attention by Évariste Galois in his seminal work on the solvability of algebraic equations in the nineteenth century. Galois showed how to associate a finite group with every polynomial and proved the remarkable result that a polynomial equation can be solved using a formula involving multiplication, division, addition, subtraction and the extraction of roots precisely when the associated finite group has this property. For this reason, finite groups of this type were called solvable (or soluble).

However, solvable groups turned out to have an importance way beyond their connection with solving polynomial equations.

Kurt Hirsch initiated the study of polycylic groups in the 1930s [?]. He showed that the following conditions on a group $$G$$ are equivalent:

1. $$G$$ is polycyclic.

2. $$G$$ has a finite series whose factors are finitely generated abelian groups.

3. $$G$$ is a solvable group that satisfies the maximal condition for subgroups. (A group is said to satisfy the maximal condition for subgroups when every non-empty set of subgroups has a maximal element, in the partial ordering of set-theoretic inclusion. This is known to be equivalent to the ascending chain condition on subgroups, and also equivalent to the condition that every subgroup is finitely generated.)

Chains of subgroups of this type are important enough to warrant some terminology, though unfortunately different authors often use the same terms somewhat differently. A chain of subgroups that extends from the trivial subgroup to the whole group and has the additional property that each member in the chain is normal in the next is sometimes called a series (or sometimes a normal series or a subnormal series). Note that it is not assumed that each of these subgroups is normal in the whole group, just that it is normal in the next one. This is enough to allow us to construct a quotient group from each pair of neighboring subgroups (as we did above with the groups $$G_i/G_{i - 1}$$). These quotient groups are referred to as the factors of the series. Properties of groups formulated in terms of series and their factors are widely used in group theory (to get some idea of the reason for their importance, refer to the article Composition series). A series whose factors all have a property $$P$$ is sometimes called a $$P$$ series. A group that has a $$P$$ series is called a poly-$$P$$ group.

The term series is often applied to a chain of subgroups like this that extends from the trivial subgroup to the whole group and has the additional property that each subgroup in the chain is normal in the next. (The terminology varies&mdash;sometimes these are called normal series or subnormal series. Note that it is not assumed that all the subgroups are normal in the whole group.) The normality condition means that we can associate a quotient group with each pair of neighbouring subgroups (in the series displayed above, these are the groups $$G_i/G_{i - 1}$$). These quotient groups are referred to as the factors of the series. Properties of groups formulated in terms of series and their factors are widely used in group theory (to get some idea of the reason for their importance, refer to the article Composition series).

The above definition of 'polycyclic' is an instance of a more general usage of the prefix 'poly' to define a group-theoretical properties via series. If, instead of assuming the factors in the series above were cyclic, we had assumed that the they all satisfy some property $$P$$, then the group $$G$$ would, in this usage, be referred to as a ‘poly-$$P$$’ group. For example, a 'polyabelian' group is a group that has a finite series with abelian factors, which amounts precisely to the concept of a solvable group (also known as 'soluble' group), according to one of the standard definitions of solvability. Another important class arises if we take $$P$$ to be 'cyclic or finite'. The class of poly-(cyclic or finite) groups

A polycyclic group is polyabelian (and therefore solvable) because any series with cyclic factors is also a series with abelian factors. In fact, for finite groups there is no distinction between polycyclic and solvable groups. This is because it can be shown (REF) that if a finite group has a series with abelian factors, then by introducing extra terms into the series, if necessary, one can always obtain a series with cyclic factors. However, for infinite groups this is not, in general, possible. The class of polycyclic groups may therefore be regarded as a natural generalisation of the class of finite solvable groups and it is natural to expect that, by avoiding some of the complexities that may arise for infinite groups, one might

History
Polycyclic groups were first studied in detail in the 1930s by Kurt Hirsch (though Hirsch did not himself use the word polycyclic). Hirsch showed (REFS) that the three following conditions were equivalent for a group $$G$$:

1. $$G$$ is polycyclic.

2. $$G$$ has a finite series whose factors are finitely generated abelian groups.

3. $$G$$ is a solvable group that satisfies the maximal condition for subgroups. (A group is said to satisfy the maximal condition for subgroups when every non-empty set of subgroups has a maximal element, in the partial ordering of set-theoretic inclusion. This is known to be equivalent to the ascending chain condition on subgroups, and also equivalent to the condition that every subgroup is finitely generated.)

Hirsch successfully extended some results from finite group theory, focusing on solvable groups, which can be viewed as built from abelian groups through a finite number of group extensions, and imposing conditions to ensure that these constituent abelian groups were of types for which we have detailed structural information.