Group Theory

9 Composition series and Jordan Hölder

For any finite group \(G\), a

normal series is a sequence \(G= G_0 \triangleright G_1 \triangleright G_2 \dots \{1\}\) The \(G_i/G_{i+1}\) are called the factors. A normal series can be refined if there is an inclusion of it into another where all the inclusions commute. This puts a partial ordering on normal series, and maximal elements are called composition series. The factors in a composition series must be simple, or else it could be further refined.

It is not hard to see that a composition series is not unique. \(C_6\) for example has two: \(C_6 \triangleright C_3 \triangleright \{1\}, C_6 \triangleright C_2 \triangleright \{1\}\). Even though these are not the same, the length is the same, and the composition factors are the same up to permutation. In fact, this is true in general.

Lemma 9.1. If we have two distinct normal subgroups \(H,H' \triangleleft G\) with simple quotient, then \(H'' = H\cap H'\) is normal in \(H',H\) with \(H'/H'' = G/H, H/H'' = G/H'\).

Proof. The natural inclusion \(H' \to G/H\) has kernel \(H''\). The image is a nontrivial normal subgroup as \(H \neq H'\), so by simplicity we are done. □

Theorem 9.2 (Jordan Hölder). Any two composition series of a finite group \(G\) are equivalent.

Proof. We can make an abstract rewriting system where our objects are pairs \((H,\sigma )\) of subgroups \(H\) of \(G\) and a formal linear combination of simple groups \(\sigma \). Our relation \(\to \) is given by \((H,\sigma ) \to (H',\sigma ')\) if \(H' \triangleleft H\), \(H/H'\) is simple, and \(\sigma ' = \sigma + H/H'\). By the previous lemma, this satisfies the conditions of the Diamond Lemma, so there is a unique normal form, proving the theorem. □

Note that this proof works in much greater generality. For example, it works for modules that are both Artinian and Noetherian (these are exactly modules with finite composition series). The theorem itself holds in even greater generality: for infinite ascending composition series of arbitrary groups for example.