# Visualising composition series

I’ve found visualising composition series using curved exact sequences to be very helpful. The notation could be used to condese elementary theorems about solvable groups into a single picture. This post follows naturally from visualising the isomorphism theorems.

## Composition series

The blue, red and green lines represent exact sequences. You could see that the fourth row are the composition factors.

## Elementary theorems about solvable groups

You could find the full proofs of these in a standard group theory textbook.

**Theorem**: If \(N\) normal subgroup of \(G\), and both \(G\) and \(G/N\) are solvable, then so is \(G\)

**Proof:** The idea is that the composition factors of \(G\) are those of \(N\) and \(G/N\) put together.

**Theorem**: If \(G\) is solvable, then any subgroup \(H\) of \(G\) is solvable

**Proof:** The idea is to let \(H_i = H \cap G_i\) where \(G_i\) is the composition series for \(G\).