---
title: p-Groups, Nilpotent, and Solvable Groups
module: Products and Group Structure
moduleNumber: 5
lessonNumber: 3
order: 503
summary: >
  Finite p-groups have nontrivial center, and iterating the center upward builds
  the nilpotent groups, which decompose as the direct product of their Sylow subgroups.
  Iterating the commutator downward builds the solvable groups, whose factors are
  abelian. The chain cyclic, abelian, nilpotent, solvable orders these classes,
  and A_5 breaks the last link.
topics: [Products and Group Structure]
sources:
  - book: Dummit & Foote
    ref: "Ch. 6 — Further Topics in Group Theory; §6.1 p-groups, Nilpotent Groups, and Solvable Groups"
  - book: Judson
    ref: "Ch. 13 — The Structure of Groups; §13.2 Solvable Groups"
draft: false
---

Between the abelian groups and the general finite group lie two structured
classes. **Nilpotent** groups are built by stacking centers upward; they behave
almost like abelian groups and split into Sylow pieces. **Solvable** groups are
built by stripping commutators downward; they are looser, but their abelian
layers are what makes polynomial equations solvable by radicals. Both classes
rest on one fact about prime-power groups.

## The center of a p-group

A finite **$p$-group** is a group of order $p^a$. Its defining structural
property is that its center is never trivial.

> **Theorem (Properties of finite $p$-groups).** Let $P$ have order $p^a$ with
> $a \geq 1$. Then:
>
> - $Z(P) \neq 1$: the center is nontrivial.
> - Every nontrivial normal subgroup meets the center: $H \cap Z(P) \neq 1$ for
>   $1 \neq H \trianglelefteq P$.
> - $P$ has a normal subgroup of order $p^b$ for every $0 \leq b \leq a$.
> - Every proper subgroup is properly contained in its normalizer: $H <
>   N_P(H)$.
> - Every maximal subgroup has index $p$ and is normal.

All five rest on the [class
equation](/abstract-algebra/group-actions-and-sylow/conjugation-and-the-class-equation).[^df-thm1]
For the first, partition $P$ into conjugacy classes: $|P| = |Z(P)| + \sum |P :
C_P(a_i)|$, summed over noncentral class representatives. Every index $|P :
C_P(a_i)|$ in the sum is a power of $p$ greater than $1$, so $p$ divides the sum
and $p$ divides $|P|$; therefore $p$ divides $|Z(P)|$, forcing $Z(P) \neq 1$.
A nontrivial center makes induction possible: quotient by $Z(P)$, apply the
inductive hypothesis, and lift the result back.

## The upper central series

Iterating "take the center" produces an ascending chain. Since $Z(P) \neq 1$,
the quotient $P/Z(P)$ is a smaller $p$-group with its own nontrivial center,
which pulls back to a larger normal subgroup of $P$, and so on.

> **Definition (Upper central series).** For any group $G$, set $Z_0(G) = 1$ and
> define $Z_{i+1}(G)$ as the preimage in $G$ of $Z(G/Z_i(G))$, so that
> $$
> Z_{i+1}(G)/Z_i(G) = Z\bigl(G/Z_i(G)\bigr).
> $$
> The chain $1 = Z_0(G) \leq Z_1(G) \leq Z_2(G) \leq \cdots$ is the **upper
> central series**. $G$ is **nilpotent** if $Z_c(G) = G$ for some $c$; the least
> such $c$ is the **nilpotence class**.

The first term $Z_1(G) = Z(G)$ is the ordinary center. Each successive term
collects the elements that become central after quotienting by the ones already
collected. An abelian group is nilpotent of class $1$, since then $G = Z(G) =
Z_1(G)$.

$$
% caption: The upper central series climbs from $1$ to $G$ by repeatedly
% adjoining the center of the quotient. A group is nilpotent exactly when this
% climb reaches $G$ in finitely many steps.
\begin{tikzpicture}[scale=1.0, font=\small,
  lv/.style={draw, black, inner sep=3pt, fill=acc!8, minimum width=20mm}]
\definecolor{acc}{HTML}{4A6FA5}
\node[lv] (z0) at (0,0)   {$Z_0 = 1$};
\node[lv] (z1) at (0,1.1) {$Z_1 = Z(G)$};
\node[lv] (z2) at (0,2.2) {$Z_2$};
\node[lv, fill=acc!16, draw=acc] (zc) at (0,3.3) {$Z_c = G$};
\draw[acc, thick, ->] (z0) -- (z1) node[midway, right, black, font=\scriptsize] {add $Z(G)$};
\draw[acc, thick, ->] (z1) -- (z2) node[midway, right, black, font=\scriptsize] {add next center};
\draw[acc, thick, ->] (z2) -- (zc) node[midway, right, black, font=\scriptsize] {climb continues};
\end{tikzpicture}
$$

Every finite $p$-group is nilpotent: each quotient $P/Z_i(P)$ is a nontrivial
$p$-group with nontrivial center, so the series strictly ascends until it reaches
$P$. A group of order $p^a$ with $a \ge 2$ has nilpotence class at most $a-1$ (a group of order $p$ is abelian, of class $1$).[^df-prop2] For
example $D_8$ and $Q_8$ both have class $2$, and $D_{2^n}$ has class $n-1$.

## Characterizations of nilpotence

For finite groups, nilpotence has several equivalent characterizations; the most
useful is structural: a finite nilpotent group is the direct product of its
Sylow subgroups.

> **Theorem (Nilpotence criteria).** For a finite group $G$ with Sylow
> subgroups $P_1, \dots, P_s$ (one per prime), the following are equivalent:
>
> - $G$ is nilpotent;
> - every proper subgroup is proper in its normalizer, $H < N_G(H)$;
> - every Sylow subgroup is normal;
> - $G \cong P_1 \times P_2 \times \cdots \times P_s$.

The normalizer condition forces each Sylow subgroup to be self-normalizing only
when it is the whole group, so every Sylow subgroup is normal; normal Sylow
subgroups for distinct primes have coprime orders and trivial pairwise
intersection, so the [recognition
theorem](/abstract-algebra/products-and-group-structure/direct-products-and-finite-abelian-groups)
assembles them into a direct product; and a product of nilpotent $p$-groups is
nilpotent.[^df-thm3]

$$
% caption: A finite nilpotent group splits as the direct product of its Sylow
% subgroups, one per prime dividing the order. Each factor is a nilpotent
% $p$-group with nontrivial center.
\begin{tikzpicture}[scale=1.0, font=\small,
  bx/.style={draw, black, inner sep=4pt, fill=acc!10, minimum width=15mm}]
\definecolor{acc}{HTML}{4A6FA5}
\node[bx, fill=acc!16, draw=acc, minimum width=20mm] (G) at (0,1.6) {$G$ nilpotent};
\node[bx] (p1) at (-3.0,0) {$P_1$};
\node[bx] (p2) at (-1.0,0) {$P_2$};
\node[black] (dots) at (1.0,0) {$\ldots$};
\node[bx] (ps) at (3.0,0) {$P_s$};
\draw[acc, thick, ->] (G) -- (p1);
\draw[acc, thick, ->] (G) -- (p2);
\draw[acc, thick, ->] (G) -- (ps);
\node[black, font=\scriptsize] at (0,-0.75) {direct product of the Sylow subgroups};
\end{tikzpicture}
$$

The first part of the Fundamental Theorem of Finite Abelian Groups — that a
finite abelian group is the direct product of its Sylow subgroups — is the
abelian special case of this theorem, since abelian groups are nilpotent.[^df-cor4]

There is a companion descending series. The **lower central series** sets $G^1 =
[G,G]$ and $G^{i+1} = [G, G^i]$; a group is nilpotent if and only if $G^n = 1$
for some $n$, and the nilpotence class is the least such $n$.[^df-thm8] The upper
and lower series climb and descend between the same endpoints but need not agree
term by term.

### A worked central series

> **Worked example.** The upper and lower central series of $D_8 = \langle r, s
> \mid r^4 = s^2 = 1,\ s^{-1}rs = r^{-1}\rangle$, of order $8$. Its center is
> $Z(D_8) = \langle r^2\rangle$, of order $2$. In the quotient $D_8/\langle
> r^2\rangle$, the Klein four-group, every element is central, so the next term
> collects all of $D_8$: $Z_2(D_8) = D_8$. The upper central series is
> $$
> 1 \;\leq\; \langle r^2\rangle \;\leq\; D_8,
> $$
> of length $2$, so $D_8$ is nilpotent of class $2$. The lower central series
> descends in step: $G^1 = [D_8, D_8] = \langle r^2\rangle$ and $G^2 = [D_8,
> \langle r^2\rangle] = 1$, since $r^2$ is central. Both series have length $2$
> and the same endpoints, and here they pass through the same middle term
> $\langle r^2\rangle$, a coincidence special to class-$2$ groups.

The same computation generalizes: $D_{2^n}$ is nilpotent of class $n-1$, while
$D_{2n}$ for $n$ not a power of $2$ is not nilpotent, because then its Sylow
subgroups are not all normal.

### Maximal subgroups and Frattini's argument

Nilpotence is also detectable from maximal subgroups alone.

> **Proposition (Maximal subgroups of a nilpotent group).** A finite group is
> nilpotent if and only if every maximal subgroup is normal.

One direction is the normalizer condition: in a nilpotent group $M < N_G(M)$, so
maximality forces $N_G(M) = G$ and $M \trianglelefteq G$. The converse uses
**Frattini's argument**: if $P$ is a Sylow subgroup of a normal subgroup $H
\trianglelefteq G$, then $G = H\,N_G(P)$.[^df-frattini] Applying it, if some
Sylow $p$-subgroup $P$ were not normal, a maximal subgroup $M$ containing $N_G(P)$
would be normal, and Frattini gives $G = M\,N_G(P) = M$, a contradiction. So
every Sylow subgroup is normal and $G$ is nilpotent. In a nilpotent group, then,
every maximal subgroup has prime index, tightening the general $p$-group fact
that maximal subgroups have index $p$.

## The derived series and solvability

Solvability weakens nilpotence by iterating a different commutator. Instead of
$[G, G^i]$, take the commutator of each term **with itself**.

> **Definition (Derived series).** For any group $G$, set $G^{(0)} = G$ and
> $G^{(i+1)} = [G^{(i)}, G^{(i)}]$. The chain
> $$
> G = G^{(0)} \trianglerighteq G^{(1)} \trianglerighteq G^{(2)} \trianglerighteq
> \cdots
> $$
> is the **derived series**. $G$ is **solvable** if $G^{(n)} = 1$ for some $n$;
> the least such $n$ is the **solvable length**.

Each $G^{(i)}$ is characteristic in $G$, and each quotient $G^{(i)}/G^{(i+1)}$ is
abelian, being a group modulo its own commutator subgroup.[^df-thm9] This
matches the original definition of solvability — a chain of subgroups with
abelian successive quotients — because $G/H$ is abelian exactly when $G' \leq H$,
so any abelian-quotient chain refines to the derived series.

$$
% caption: The derived series descends by taking commutator subgroups, and each
% successive quotient $G^{(i)}/G^{(i+1)}$ is abelian. Solvability means the
% series reaches the identity.
\begin{tikzpicture}[scale=1.0, font=\small,
  lv/.style={draw, black, inner sep=3pt, fill=acc!8, minimum width=22mm}]
\definecolor{acc}{HTML}{4A6FA5}
\node[lv, fill=acc!16, draw=acc] (g0) at (0,3.3) {$G^{(0)} = G$};
\node[lv] (g1) at (0,2.2) {$G^{(1)}$};
\node[lv] (g2) at (0,1.1) {$G^{(2)}$};
\node[lv] (gn) at (0,0)   {$G^{(n)} = 1$};
\draw[acc, thick, ->] (g0) -- (g1) node[midway, right, black, font=\scriptsize] {abelian quotient};
\draw[acc, thick, ->] (g1) -- (g2) node[midway, right, black, font=\scriptsize] {abelian quotient};
\draw[acc, thick, ->] (g2) -- (gn) node[midway, right, black, font=\scriptsize] {reaches $1$};
\end{tikzpicture}
$$

Solvability is closed under three operations.[^df-prop10]

- **Subgroups.** $H \leq G$ solvable implies $H$ solvable, since $H^{(i)} \leq
  G^{(i)}$.
- **Quotients.** A homomorphic image of a solvable group is solvable, since
  $\varphi(G^{(i)}) = \varphi(G)^{(i)}$.
- **Extensions.** If $N \trianglelefteq G$ with both $N$ and $G/N$ solvable, then
  $G$ is solvable.

## The hierarchy of structure

The classes nest in a strict chain, each obtained by loosening the last. Cyclic
groups are abelian; abelian groups are nilpotent of class $1$; nilpotent groups
are solvable, taking the upper central series as an abelian-quotient chain; and
solvable groups sit inside all groups.

$$
% caption: The strict containment cyclic $\subset$ abelian $\subset$ nilpotent
% $\subset$ solvable $\subset$ all groups. Each inclusion is proper, witnessed by
% a group in the annulus just outside the smaller class.
\begin{tikzpicture}[scale=1.0, font=\small]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, fill=acc!4] (0,0) ellipse (4.0 and 2.5);
\draw[black, fill=acc!7] (-0.5,0) ellipse (3.1 and 2.0);
\draw[black, fill=acc!10] (-1.0,0) ellipse (2.2 and 1.5);
\draw[black, fill=acc!13] (-1.4,0) ellipse (1.35 and 1.0);
\draw[acc, fill=acc!18] (-1.7,0) ellipse (0.7 and 0.55);
\node[acc, font=\scriptsize] at (-1.7,0) {cyclic};
\node[black, font=\scriptsize] at (-1.05,0.9) {abelian};
\node[black, font=\scriptsize] at (-0.35,1.4) {nilpotent};
\node[black, font=\scriptsize] at (0.5,1.9) {solvable};
\node[black, font=\scriptsize] at (2.7,2.15) {all groups};
% witnesses
\node[black, font=\scriptsize] at (2.5,0.4) {$S_3$ solvable};
\node[black, font=\scriptsize] at (2.7,-0.4) {not nilpotent};
\node[black, font=\scriptsize] at (0.2,-2.15) {$A_5$ outside all};
\end{tikzpicture}
$$

Each inclusion is proper.

> **Worked example.** $S_3$ is solvable but not nilpotent. The derived series is
> $$
> S_3 \;\trianglerighteq\; A_3 \;\trianglerighteq\; 1,
> $$
> since $S_3' = [S_3, S_3] = A_3$ and $A_3 \cong \mathbb{Z}_3$ is abelian, so
> $A_3' = 1$. The successive quotients $S_3/A_3 \cong \mathbb{Z}_2$ and $A_3
> \cong \mathbb{Z}_3$ are abelian, so $S_3$ is solvable of length $2$. It is not
> nilpotent: $Z(S_3) = 1$, so every upper-central term $Z_n(S_3) = 1$ and the
> climb never reaches $S_3$. The lower central series stabilizes at $[S_3, S_3] =
> A_3 \neq 1$, so for this non-nilpotent group the upper and lower series have
> different lengths.

## Solvability and the quintic

Solvability is the class where composition series have abelian factors.

> **Definition (Solvable via composition series).** A finite group $G$ is
> solvable if it has a subnormal series with abelian factor groups; equivalently,
> the simple factors in any [composition
> series](/abstract-algebra/subgroups-and-quotients/composition-series-and-the-alternating-group)
> are all cyclic of prime order.

> **Worked example.** $S_4$ is solvable. Its derived series is
> $$
> S_4 \;\trianglerighteq\; A_4 \;\trianglerighteq\; V \;\trianglerighteq\; 1,
> $$
> where $V = \{1, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ is the Klein four
> normal subgroup: $S_4' = A_4$, $A_4' = V$, and $V' = 1$ since $V$ is abelian.
> The factors $S_4/A_4 \cong \mathbb{Z}_2$, $A_4/V \cong \mathbb{Z}_3$, and $V
> \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ are all abelian, so $S_4$ is solvable of
> length $3$.[^ju-solv]

The pattern breaks at $A_5$. For $n \geq 5$ the group $A_n$ is simple and
non-abelian, so the composition series $S_n \trianglerighteq A_n \trianglerighteq
1$ has the non-abelian factor $A_n$, and $S_n$ is not solvable.

$$
% caption: $S_4$ is solvable through the chain $S_4 \trianglerighteq A_4
% \trianglerighteq V \trianglerighteq 1$ with abelian factors; $A_5$ is simple
% and non-abelian, so no such abelian-factor chain descends from $S_5$.
\begin{tikzpicture}[scale=1.0, font=\small,
  nd/.style={draw, black, inner sep=3pt, fill=acc!10, minimum width=13mm}]
\definecolor{acc}{HTML}{4A6FA5}
% solvable ladder S4
\node[nd] (s4) at (0,3.0) {$S_4$};
\node[nd] (a4) at (0,2.0) {$A_4$};
\node[nd] (v)  at (0,1.0) {$V$};
\node[nd] (e)  at (0,0.0) {$1$};
\draw[acc, thick, ->] (s4) -- (a4) node[midway, right, black, font=\scriptsize] {$\mathbb{Z}_2$};
\draw[acc, thick, ->] (a4) -- (v)  node[midway, right, black, font=\scriptsize] {$\mathbb{Z}_3$};
\draw[acc, thick, ->] (v)  -- (e)  node[midway, right, black, font=\scriptsize] {$(\mathbb{Z}_2)^2$};
\node[black, font=\scriptsize] at (0,-0.55) {$S_4$ solvable};
% non-solvable S5
\node[nd] (s5) at (5.0,3.0) {$S_5$};
\node[nd, fill=acc!18, draw=acc] (a5) at (5.0,1.6) {$A_5$ simple};
\node[nd] (e2) at (5.0,0.0) {$1$};
\draw[acc, thick, ->] (s5) -- (a5) node[midway, right, black, font=\scriptsize] {$\mathbb{Z}_2$};
\draw[acc, thick, ->] (a5) -- (e2) node[midway, right, acc, font=\scriptsize] {non-abelian};
\node[black, font=\scriptsize] at (5.0,-0.55) {$S_5$ not solvable};
\end{tikzpicture}
$$

This is the group-theoretic content behind the [insolvability of the
quintic](/abstract-algebra/galois-theory/solvability-by-radicals-and-the-quintic):
a polynomial is solvable by radicals exactly when its Galois group is solvable,
and the general quintic has Galois group $S_5$, which is not. Several deep
sufficient conditions for solvability round out the theory — a group of order
$p^a q^b$ is solvable (Burnside), and every group of odd order is solvable
(Feit–Thompson) — but the relevant fact for Galois theory is the sharp boundary
at $A_5$.[^df-thm11]

## Summary of the two constructions

| | Nilpotent | Solvable |
| --- | --- | --- |
| Built by | upper central series (climb) | derived series (descend) |
| Terminates when | $Z_c(G) = G$ | $G^{(n)} = 1$ |
| Factor groups | central extensions | abelian |
| Finite characterization | product of Sylow subgroups | prime-cyclic composition factors |
| Smallest failure | $S_3$ (solvable, not nilpotent) | $A_5$ (not solvable) |
| Contains | all finite $p$-groups | all nilpotent groups |

Nilpotent groups are close enough to abelian that Sylow subgroups separate
cleanly; solvable groups keep only the abelian-layer structure that Galois theory
uses. Both rest on the single fact that a $p$-group has a nontrivial center.

[^df-thm1]: **Dummit & Foote**, _Abstract Algebra_, §6.1, Theorem 1: the five structural properties of finite $p$-groups, all consequences of the class equation, including $Z(P) \neq 1$ and normal subgroups of every intermediate order.
[^df-prop2]: **Dummit & Foote**, §6.1, Proposition 2: a group of order $p^a$ is nilpotent of class at most $a-1$; $D_{2^n}$ has class $n-1$.
[^df-thm3]: **Dummit & Foote**, §6.1, Theorem 3: the equivalence of nilpotence with the normalizer condition, normality of all Sylow subgroups, and the direct product of Sylow subgroups.
[^df-cor4]: **Dummit & Foote**, §6.1, Corollary 4: a finite abelian group is the direct product of its Sylow subgroups, recovered as the abelian case of Theorem 3.
[^df-thm8]: **Dummit & Foote**, §6.1, Theorem 8: $G$ is nilpotent iff the lower central series reaches $1$, with the class equal to the length; the relation $Z_i(G) \leq G^{c-i-1} \leq Z_{i+1}(G)$.
[^df-frattini]: **Dummit & Foote**, §6.1, Proposition 6 (Frattini's Argument) and Proposition 7: if $P$ is a Sylow subgroup of $H \trianglelefteq G$ then $G = H\,N_G(P)$; a finite group is nilpotent iff every maximal subgroup is normal.
[^df-thm9]: **Dummit & Foote**, §6.1, Theorem 9: $G$ is solvable iff $G^{(n)} = 1$ for some $n$; the derived series is the shortest abelian-quotient series and consists of characteristic subgroups.
[^df-prop10]: **Dummit & Foote**, §6.1, Proposition 10: subgroups and quotients of solvable groups are solvable, and an extension of a solvable group by a solvable group is solvable.
[^ju-solv]: **Judson**, _Abstract Algebra_, §13.2 — Solvable Groups: the composition-series definition of solvability, $S_4$ solvable via the chain through $A_4$ and $V$, and $S_n$ not solvable for $n \geq 5$.
[^df-thm11]: **Dummit & Foote**, §6.1, Theorem 11: sufficient conditions for solvability, including Burnside's $p^a q^b$ theorem and the Feit–Thompson odd-order theorem.