Difference between revisions of "Composition series"
(New page: A '''composition series''' is a way of describing a group. == Definition == A composition series of a group <math>G</math> with idenitity <math>e</math> is a finite sequence <math>(G...) |
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* [[Jordan-Hölder series]] | * [[Jordan-Hölder series]] | ||
* [[Zassenhaus's Lemma]] | * [[Zassenhaus's Lemma]] | ||
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* [[Jordan-Hölder Theorem]] | * [[Jordan-Hölder Theorem]] | ||
Latest revision as of 20:03, 10 May 2008
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://latex.artofproblemsolving.com/6/f/4/6f459a163e3ae43553cbefbd1f0ac5b3160e7e92.png)
, 
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://latex.artofproblemsolving.com/b/5/9/b59d10a732c011f075f1b58998bd3c86f0b28526.png)
such that and 
are isomorphic for all integers .
Resources
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