Polycyclic group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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History

Origin of the concept

Hirsch, in the years 1938-1954, obtained results on polycylic groups, but he used the term S-group for them.

Origin of the term

The term polycyclic group was first used by Hall in 1954. It is now the commonly accepted term.

Definition

A group is said to be polycyclic if it has a subnormal series (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.

Definition with symbols

A group $G$ is said to be polycyclic if there exists a series of subgroups:

$1 = H_{0} ◃ H_{1} ◃ H_{2} ◃ \dots ◃ H_{n} = G$

where each $H_{i + 1} / H_{i}$ is cyclic.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Any subgroup of a polycyclic group is polycyclic. The subnormal serise for the subgroup is obtained by intersecting with it the subnormal series of the whole group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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Any quotient of a polycyclic group is polycyclic. The subnormal series for the quotient is obtained by taking the image of the subnormal series for the whole group, via the quotient map.

Template:P-group

A direct product of polycyclic groups is polycyclic.