Group Theory

David Speyer's notes on Jordan-Holder for finite groups: PDF

Idea is, look at a normal lattice $G$ containing large normal subgroups $N,N'$ containing $N\cap N'$ (intersection of normals is normal), applying induction to these subgroups. The point is the map $(G\cap N)/(N'\cap N) \to G/N'$ is always injective, so by simplicity of $G/N'$, it's either trivial or an isomorphism. Put another way, intersecting a simple filtration with $N$ is well-behaved: the successive quotients will either be preserved or eliminated.

In the abelian case or with modules (either over a commutative or non-commutative ring, former being useful in commutative algebra, algebraic number theory, and algebraic geometry and latter being useful in general representation theory), we can also go the other way (up instead of down): let $X = N+N'$ for small submodules $N,N'$, and apply induction to the quotients like $M/N$, $M/N'$, and $M/X$. Both intersection with $X$ and addition by $X$ are well-behaved: $$0 \to (M\cap X) / (N\cap X) \to M/N \to (M+X)/(N+X) \to 0$$ is short exact. The previous (group) approach with large $N,N'$ uses that $(M\cap X) / (N\cap X)$ is well-behaved; the alternative approach here uses that $(M+X)/(N+X)$ is well-behaved.