Composition series
From AoPSWiki
A composition series is a way of describing a group.
Definition
A composition series of a group 
with idenitity 
is a finite sequence 
of subgroups of such that 
, 
, and for each integer ![i \in [0,n-1]](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/b/9/e/b9ef231022cf009a80cb4ac041c905696f4b9997.gif)
, 
is a normal subgroup of 
.
The quotient groups 
are called the quotients of the series. We call a composition series 
finer than a composition series 
if the terms of are taken from the terms of . Note, however, that in general, a composition series with some terms omitted is no longer a composition series, since in general if 
is a normal subgroup of 
and is a normal subgroup of , then is not necessarily a normal subgroup of .
Two composition series and 
(of not necessarily identical groups and ) are considered equivalent if 
, and there is a permutation 
of the integers in ![[0,n-1]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/5/0/8/50843be249065f561612efbc08eef98aba77853b.gif)
such that and 
are isomorphic for all integers .
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