A composition series of a finite group 
of order greater than one is
a finite series of subgroups
such that
 |
where 
is a maximal proper normal subgroup of 
and 
means that 
is a normal subgroup of
. A composition series is therefore a normal
series without repetition whose factors are all simple
(Scott 1987, p. 36). Every finite group of order greater than one possesses a composition series.
The quotient groups 
, 
, ..., 
, 
are called composition quotient groups.
For a module 
, a composition series is a finite strictly descending chain of submodules 
whose successive quotient modules are irreducible
modules. Equivalently, no further submodule can
be inserted between consecutive terms. Its length 
is the module length (Atiyah
and Macdonald 1969, pp. 76-77).
