proof of the Jordan Hölder decomposition theorem

Let |G|=N. We first prove existence, using inductionMathworldPlanetmath on N. If N=1 (or, more generally, if G is simple) the result is clear. Now suppose G is not simple. Choose a maximal proper normal subgroup G1 of G. Then G1 has a Jordan–Hölder decomposition by induction, which produces a Jordan–Hölder decomposition for G.

To prove uniqueness, we use induction on the length n of the decomposition series. If n=1 then G is simple and we are done. For n>1, suppose that




are two decompositions of G. If G1=G1 then we’re done (apply the induction hypothesis to G1), so assume G1G1. Set H:=G1G1 and choose a decomposition series


for H. By the second isomorphism theorem, G1/H=G1G1/G1=G/G1 (the last equality is because G1G1 is a normal subgroupMathworldPlanetmath of G properly containing G1). In particular, H is a normal subgroup of G1 with simple quotientPlanetmathPlanetmath. But then




are two decomposition series for G1, and hence have the same simple quotients by the induction hypothesis; likewise for the G1 series. Therefore n=m. Moreover, since G/G1=G1/H and G/G1=G1/H (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 theoremMathworldPlanetmath
Canonical name ProofOfTheJordanHolderDecompositionTheorem
Date of creation 2013-03-22 12:08:49
Last modified on 2013-03-22 12:08:49
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Proof
Classification msc 20E22