Simple Group Decomposition

If is a finite group, then it has a composition series:

where the Quotient Group is simple for each .

Proof

By induction on .
If is simple, done.
Otherwise find a proper Normal Subgroup of maximal order .
Suppose has a normal subgroup .
Then there is a normal subgroup of s.t. is normal in .
But then contains so, as is maximal, has to be ,
but then hence induct.