English
In a simple group, the derived series stabilizes immediately at the commutator; i.e., D_{n+1} = [G,G] for all n or else equals commutator chain; specifically, D_{n+1} = commutator G for simple G.
Русский
В простой группе производящая серия сводится к коммутатору; т.е. D_{n+1} = [G,G] для всех n или эквивалентно коммутаторной цепи; в частности, D_{n+1} = commutator G.
LaTeX
$$derivedSeries G (n.succ) = commutator G$$
Lean4
theorem derivedSeries_succ {n : ℕ} : derivedSeries G n.succ = commutator G := by
induction n with
| zero => exact derivedSeries_one G
| succ n ih =>
rw [_root_.derivedSeries_succ, ih, _root_.commutator]
rcases (commutator_normal (⊤ : Subgroup G) (⊤ : Subgroup G)).eq_bot_or_eq_top with h | h
· rw [h, commutator_bot_left]
· rwa [h]
