Nilpotent Group

Statement

The lower central series of $G$ is $\gamma_0(G) = G$, $\gamma_{k+1}(G) = [G, \gamma_k(G)]$. A group $G$ is nilpotent if $\gamma_n(G) = \{e\}$ for some $n$, called the nilpotency class.

Equivalently, the upper central series $1 = Z_0 \leq Z_1 \leq \cdots$ (where $Z_{i+1}/Z_i = Z(G/Z_i)$) reaches $G$ in $n$ steps.

Key fact: every finite $p$-group (group of order $p^k$) is nilpotent.

Visualization

Lower central series of $D_4$ vs $S_3$:

$D_4 = \langle r, s \mid r^4 = s^2 = e,\; srs^{-1} = r^{-1}\rangle$ (dihedral group of order 8):

D_4       (order 8, nilpotency class 2)
 |   [D_4, D_4] = {e, r^2}  (center, order 2)
{e,r^2}   (order 2, central)
 |   [{e,r^2}, D_4] = {e}
{e}

$S_3 = \langle (12),(123)\rangle$ (symmetric group of order 6):

S_3       (order 6, NOT nilpotent)
 |   [S_3, S_3] = A_3 = {e,(123),(132)}
A_3       (order 3)
 |   [A_3, S_3] = A_3  (stabilizes! never reaches {e})
A_3       ...

$S_3$ is not nilpotent because $[A_3, S_3] = A_3 \neq \{e\}$. The lower central series stabilizes above $\{e\}$.

$p$-groups are nilpotent: For $G = \mathbb{Z}/4 \times \mathbb{Z}/2$ (order $2^3$):

| Step $k$ | $Z_k$ (upper central series) | $|Z_k|$ | |---|---|---| | 0 | $\{0\}$ | 1 | | 1 | $G$ | 8 |

$Z_1 = G$ immediately — $G$ is abelian, so it coincides with its own center, giving nilpotency class 1.

Proof Sketch

1. $p$-groups have non-trivial center. By the class equation, $|G| \equiv |Z(G)| \pmod{p}$. Since $p \mid |G|$, we get $p \mid |Z(G)|$, so $Z(G) \neq \{e\}$.
2. Induct on $|G|$. The center $Z(G)$ is non-trivial by step 1. The quotient $G/Z(G)$ is a $p$-group of smaller order, hence nilpotent by induction.
3. Lift nilpotency. The preimage of the upper central series of $G/Z(G)$ gives the upper central series of $G$, reaching $G$ in one extra step.
4. Nilpotent implies solvable. Every step $G_k/G_{k+1}$ in the upper central series is abelian (it lies in a center), so the abelian quotient chain certifies solvability.

Connections

Nilpotent groups are the "tamest" non-abelian groups. The class equation driving step 1 is Class EquationClass Equation$$|G| = |Z(G)| + \sum_{\text{non-central}} |\mathrm{cl}(g)|$$The order of a finite group equals the size of its center plus the sum of sizes of nontrivial conjugacy classes.Read more →. Nilpotent groups are always solvable — see Solvable GroupSolvable Group$$G = G^{(0)} \supset G^{(1)} \supset \cdots \supset G^{(n)} = \{e\}$$A group is solvable if its derived series reaches the trivial group in finitely many steps.Read more → for the comparison. Finite $p$-groups appear throughout Sylow TheoremsSylow Theorems$$p^k \mid |G| \implies \exists H \leq G,\; |H| = p^k$$Prime-power subgroups always exist, are conjugate, and their count is constrainedRead more →, where Sylow subgroups of nilpotent groups are always normal.

/-- Every finite p-group is nilpotent. Mathlib:
    `IsPGroup.isNilpotent` in GroupTheory.Nilpotent. -/
theorem pGroup_isNilpotent {G : Type*} [Group G] [Finite G] {p : ℕ}
    [Fact p.Prime] (h : IsPGroup p G) : IsNilpotent G :=
  IsPGroup.isNilpotent h

/-- Abelian groups are nilpotent (class 1): their lower central series reaches
    {e} in one step because [G, G] = {e}. Mathlib's instance: -/
instance abelian_nilpotent {G : Type*} [CommGroup G] : IsNilpotent G :=
  CommGroup.isNilpotent

/-- Subgroups of nilpotent groups are nilpotent. -/
example {G : Type*} [Group G] [IsNilpotent G] (H : Subgroup G) :
    IsNilpotent H :=
  Subgroup.isNilpotent H