In
mathematics, especially in the fields of
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 ...
and
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
, a central series is a kind of
normal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
of
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s or
Lie subalgebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s, expressing the idea that the
commutator is nearly trivial. For
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, the existence of a central series means it is a
nilpotent group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with .
Intui ...
; for
matrix rings (considered as Lie algrebras), it means that in some basis the ring consists entirely of
upper triangular
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal ar ...
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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
. A related but distinct construction is the
derived series
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group.
Th ...
, which terminates in the trivial subgroup whenever the group is
solvable.
Definition
A central series is a sequence of subgroups
:
such that the successive quotients are
central; that is,
, where