Group Theory

5 Nilpotent Groups

Definition 5.1. A group \(G\) is nilpotent if there is a sequence (called a central series) \({1}=G_0\triangleleft G_1,\dots ,\triangleleft G_n = G\) with each \(G_i\) central (ie. in the center) in \(G/G_{i-1}\).

The length of the shortest central series, \(n\) is called the nilpotent class of \(G\). For example, abelian groups are those whose nilpotent class is \(1\). The nilpotent class measures how far away from being abelian \(G\) is.

There are two canonical “extremal" central series for a group, the upper and lower central series.

Definition 5.2. The lower central series of a group \(G\) is defined by \(G_1 = G, G_{i+1} = [G,G_i]\).

Since \(c[g,h]c^{-1} = [cgc^{-1},chc^{-1}]\), by induction we see \(G_i\) is normal, and it is clear that for any normal subgroup \(H \triangleleft G\), \(H\) is central in \(G/[G,H]\).

Definition 5.3. The upper central series of a group \(G\) is defined by \(G^1 = \{1\}, G^{i+1}/G^i = Z(G/G^i)\).

It is clear that if either some \(G_i = \{1\}\) or some \(G^i = G\), then \(G\) is nilpotent.

Moreover we can observe the following two basic results, which explain why the lower and upper central series are extremal.

Proposition 5.4. If \(1=H^0\triangleleft ,\dots ,\triangleleft H^n = G\) is a central series, then \(H^i\subset G^i\).

Proof. This follows from induction since if \(H^i \subset G^i\), then \(H^{i+1}\subset Z(G/H_i) \subset Z(G/G_i) = G^{i+1}\). □

Proposition 5.5. If \(G=H_0\triangleright ,\dots ,\triangleright H_n = \{1\}\) is a central series, then \(H_i \supset G_i\).

Proof. This follows from induction and the fact that if \(H_i \supset G_i\), then \(H_{i+1} \supset [G,H_{i}] \supset [G,G_i] = G_{i+1}\). □

Corollary 5.6. For a nilpotent group, the lengths of the upper and lower central series are the nilpotent classes of \(G\). In particular a group is nilpotent iff the two central series stabilize at \(G\) and \(\{1\}\).

Proof. The previous two propositions imply they are minimal central series. □

Proposition 5.7. Nilpotent groups are solvable.

Proof. Let \(G^{i+1}/G^{i}\) is abelian as it is the center of \(G/G^{i}\). □

Lemma 5.8. If \(g,h\) are relatively prime order in a nilpotent group \(G\), then they commute.

Proof. We can induct on nilpotent class. It then suffices to show that if this holds for \(G/Z(G)\), then it holds for \(G\). If \(g,h\) are order \(a,b, (a,b) = 1\), then \([g,h]\) is in \(Z(G)\) by induction. By induction again, we have \([g^n,h] = g^{n-1}[g,h]hg^{1-n}h^{-1}=[g,h]^n\), so we see that the order of \([g,h]\) divides that of \(g,h\), so \([g,h]=1\). □

Theorem 5.9. The finite nilpotent groups are products of \(p\)-groups.

Proof. Finite \(p\)-groups have nontrivial center by the class equation, and since quotients of \(p\)-groups are \(p\)-groups, they are nilpotent by induction. For the converse, the previous lemma shows that all the \(p\)-Sylows for different \(p\)s commute with each other, so by Lemma 2.4 we are done. □