proof of the Jordan Hölder decomposition theorem

Let $|G|=N$. We first prove existence, using induction 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 $G_1$ of $G$. Then $G_1$ 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

$$G\supset G_1\supset G_2\supset \mathrm{\cdots }\supset G_n=\{1\}$$

and

$$G\supset G_1^{\prime }\supset G_2^{\prime }\supset \mathrm{\cdots }\supset G_m^{\prime }=\{1\}$$

are two decompositions of $G$. If $G_1=G_1^{\prime }$ then we’re done (apply the induction hypothesis to $G_1$), so assume $G_1\ne G_1^{\prime }$. Set $H:=G_1\cap G_1^{\prime }$ and choose a decomposition series

$$H\supset H_1\supset \mathrm{\cdots }\supset H_k=\{1\}$$

for $H$. By the second isomorphism theorem, $G_1/H=G_1{G}_1^{\prime }/G_1^{\prime }=G/G_1^{\prime }$ (the last equality is because $G_1{G}_1^{\prime }$ is a normal subgroup of $G$ properly containing $G_1$). In particular, $H$ is a normal subgroup of $G_1$ with simple quotient. But then

$$G_1\supset G_2\supset \mathrm{\cdots }\supset G_n$$

and

$$G_1\supset H\supset \mathrm{\cdots }\supset H_k$$

are two decomposition series for $G_1$, and hence have the same simple quotients by the induction hypothesis; likewise for the $G_1^{\prime }$ series. Therefore $n=m$. Moreover, since $G/G_1=G_1^{\prime }/H$ and $G/G_1^{\prime }=G_1/H$ (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 theorem
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