Nilpotent group

A nilpotent group is a group built from successive central extensions. For finite groups it lies strictly between the abelian and solvable classes, and it can be recognized through Sylow subgroups, normalizers, maximal subgroups, or subnormality. Every finite $$p$$ -group is nilpotent, so nilpotence is the basic mechanism that turns local prime-power structure into a global direct-product decomposition. ^{[@isaacs2008]}

Definition

For a group $$G$$ , define the upper central series by

Z_0(G)=1, \qquad Z_{i+1}(G)/Z_i(G)=Z\bigl(G/Z_i(G)\bigr).

Equivalently, $Z_{i+1}(G)$ consists of the elements $$x$$ such that $[x,G]\subseteq Z_i(G)$ .

The lower central series approaches the identity from the other direction:

\gamma_1(G)=G, \qquad \gamma_{i+1}(G)=[\gamma_i(G),G].

The group has class at most $$c$$ exactly when $\gamma_{c+1}(G)=1$ . The equivalence follows by transferring the commutator inclusions

\gamma_i(G)\leq Z_{c+1-i}(G)

between a central series and the two canonical central series.

Equivalent conditions for finite groups

Main chain of implications

Assume first that $$G$$ is nilpotent and let $$H<G$$ . Choose the least $$i$$ for which $Z_i(G)\nsubseteq H$ . Then $Z_{i-1}(G)\subseteq H$ , and $[z,H]\subseteq Z_{i-1}(G)\subseteq H$ for every $z\in Z_i(G)$ . Thus $Z_i(G)\leq N_G(H)$ , and an element of $Z_i(G)\setminus H$ proves $$H<N_G(H)$$ .

The normalizer condition implies that every maximal subgroup is normal. Conversely, suppose every maximal subgroup is normal and let $$P$$ be a Sylow $$p$$ -subgroup. If $$N_G(P)<G$$ , put it inside a maximal subgroup $$M$$ . Since $M\triangleleft G$ , the Frattini argument yields $$G=N_G(P)M=M$$ , a contradiction. Hence every Sylow subgroup is normal.

If all Sylow subgroups are normal, distinct Sylow subgroups $$P$$ and $$Q$$ satisfy $[P,Q]\leq P\cap Q=1$ , and their product is all of $$G$$ . Thus $$G$$ is their direct product. Every finite $$p$$ -group has nontrivial center and is nilpotent by induction on its order, while a finite direct product of nilpotent groups is nilpotent. This completes the cycle.

Finite p-groups

Two useful strengthenings follow:

if $1<N\triangleleft P$ , then $N\cap Z(P)>1$ ;
if $$N<M$$ are normal in $$P$$ , there exists $L\triangleleft P$ with $N<L\leq M$ and $$|L:N|=p$$ .

Iterating the second statement shows that a group of order $$p^a$$ has a normal subgroup of order $$p^b$$ for every $0\leq b\leq a$ . This is the structural content developed through Theorems 1.23--1.25 and their exercises in Isaacs. ^{[@isaacs2008]}

Sylow direct-product decomposition

Write

|G|=\prod_{i=1}^{r}p_i^{a_i}.

If $$G$$ is finite nilpotent, each $P_i\in\operatorname{Syl}_{p_i}(G)$ is unique and normal, and

G=P_1\times\cdots\times P_r.

Questions about $$G$$ can therefore be treated one prime at a time. If $x=x_1\cdots x_r$ with $x_i\in P_i$ , then the order of $$x$$ is the product of the orders of the $$x_i$$ . Every subgroup also decomposes:

H=\prod_i(H\cap P_i).

It follows at once that subgroups and quotient groups of finite nilpotent groups are nilpotent.

Examples and counterexamples

Subgroups, quotients, and extensions

Nilpotence is inherited by subgroups, quotients, and finite direct products, but not by arbitrary extensions. In

1\longrightarrow C_3\longrightarrow S_3\longrightarrow C_2\longrightarrow1,

both kernel and quotient are nilpotent, while $$S_3$$ is not. A product of nilpotent subgroups need not be nilpotent either unless suitable normality assumptions are imposed.

The theorem is the precise normal-product replacement for the false extension statement. Its proof separates both normal subgroups into Sylow components and controls commutators between components of coprime order. See Fitting subgroup.

Relation to solvability and the Frattini subgroup

Every nilpotent group is solvable because a central series is a sequence of extensions by abelian groups. For a finite $$p$$ -group $$P$$ ,

\Phi(P)=P'P^p,

and $P/\Phi(P)$ is elementary abelian. Its dimension over $\mathbf F_p$ is the minimum number of generators of $$P$$ . This is the bridge from central structure to linear algebra.

Applications

Sylow theory decomposes a finite nilpotent group into prime-power factors.
Fitting theory concentrates all normal nilpotent structure in $$F(G)$$ .
Thompson's theorem says that every finite Frobenius kernel is nilpotent.
The normalizer condition forces proper subgroups to grow inside local $$p$$ -structure.
In commutator calculus, the nilpotency class bounds the length of nontrivial iterated commutators.

The finite criteria, normalizer condition, and Sylow decomposition above are independently organized around Isaacs, Chapter 1. Terminology and general definitions were cross-checked against the cited public encyclopedia revision. ^{[@isaacs2008]} ^{[@wikipedia-nilpotent]}