I came across a simple statement in finite group theory that I’m almost upset no one told me earlier. The source is Serre’s book ‘Linear Representations of Finite Groups’. Serre used this statement below to define / prove Brauer’s theorem on induced representations of finite groups, using which one proves the meromorphicity of Artin-L-functions. Here it goes.

$G$ is a finite group and $p$ is a fixed prime. An element $x$ of $G$ is called $p$-unipotent if $x$ has order a power of $p$ and $p$-regular if it’s order is prime to $p$.

Cool result:  Every element $x$ in $G$ can be uniquely written as

$x = x_u x_r;$

where

• $x_u$ is $p$-unipotent, $x_r$ is $p$-regular,
• $x_u$ and $x_r$ commute and
• they are both powers of $x$.

The proof is really easy. Just replace $G$ by the (finite) cyclic group generated by $x$!

I call it the Jordan decomposition because we have a similar decomposition for endomorphisms (among other things).

Let $V$ be a finite dimensional vector space over an algebraically closed field of characteristic zero (just in case!). Each $x \in \text{End }(V)$ can be uniquely written as

$x = x_s x_u;$

where

• $x_s$ is semisimple (diagonalizable), $x_u$ is nilpotent,
• they both commute and
• they are polynomials in $x$ without a constant term.

Pretty cool, huh!