In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 L ...
, 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
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
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 .
Intuiti ...
; for
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
s (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 class ...
. 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
Central is an adjective usually referring to being in the center of some place or (mathematical) object.
Central may also refer to:
Directions and generalised locations
* Central Africa, a region in the centre of Africa continent, also known as ...
; that is,
, where