Lower central series
The lower central series of G is γ₁ = G ⊇ γ₂ = [G, G] ⊇ γ₃ ⊇ …, with
γk+1 = [γk, G]. G is nilpotent exactly when the series
reaches the trivial subgroup; the number of steps is its nilpotency class.
Its limit γ∞(G) is the hypocenter. For a non-nilpotent G it stops
above 1 — e.g. γ∞ = G when G is perfect (G = [G, G], as
for any non-abelian simple group such as A₅).