proof of the Jordan Hölder decomposition theorem
Let . We first prove existence, using induction on . If (or, more generally, if is simple) the result is clear. Now suppose is not simple. Choose a maximal proper normal subgroup of . Then has a Jordan–Hölder decomposition by induction, which produces a Jordan–Hölder decomposition for .
To prove uniqueness, we use induction on the length of the decomposition series. If then is simple and we are done. For , suppose that
are two decompositions of . If then we’re done (apply the induction hypothesis to ), so assume . Set and choose a decomposition series
are two decomposition series for , and hence have the same simple quotients by the induction hypothesis; likewise for the series. Therefore . Moreover, since and (by the second isomorphism theorem), we have now accounted for all of the simple quotients, and shown that they are the same.
|Title||proof of the Jordan Hölder decomposition theorem|
|Date of creation||2013-03-22 12:08:49|
|Last modified on||2013-03-22 12:08:49|
|Last modified by||djao (24)|