p-Groups, Nilpotent, and Solvable Groups

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.

All five rest on the class equation.¹ 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 $Z(P)\ne 1$. 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 $Z(P)\ne 1$, 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.

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 $G=Z(G)=Z_1(G)$.

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 $1$).² 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.

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 assembles them into a direct product; and a product of nilpotent $p$-groups is nilpotent.³

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.⁴

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 $n$.⁵ The upper and lower series climb and descend between the same endpoints but need not agree term by term.

A worked central series

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.

One direction is the normalizer condition: in a nilpotent group $M<N_G(M)$, so maximality forces $N_G(M)=G$ and $M⊴G$. The converse uses Frattini's argument: if $P$ is a Sylow subgroup of a normal subgroup $H⊴G$, then $G=H{N}_G(P)$.⁶ 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.

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.⁷ This matches the original definition of solvability — a chain of subgroups with abelian successive quotients — because $G/H$ is abelian exactly when $G^{\prime }\le H$, so any abelian-quotient chain refines to the derived series.

Solvability is closed under three operations.⁸

• Subgroups. $H\le G$ solvable implies $H$ solvable, since $H^{(i)}\le G^{(i)}$.
• Quotients. A homomorphic image of a solvable group is solvable, since $\phi (G^{(i)})=\phi (G{)}^{(i)}$.
• Extensions. If $N⊴G$ 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.

Each inclusion is proper.

Solvability and the quintic

Solvability is the class where composition series have abelian factors.

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

This is the group-theoretic content behind the insolvability of 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 $A_5$.¹⁰

Summary of the two constructions

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.