Prime power order implies nilpotent

Statement

Verbal statement

Any group of prime power order is nilpotent.

Related results

• Prime power order implies not centerless: This result is the key ingredient used to prove that any group of prime power order is nilpotent.
• Prime power order implies center is normality-large: This is a stronger version of the fact that any group of prime power order is centerless.

Proof

We prove the statement by showing that it is possible to construct an upper central series for the group. The proof proceeds by induction on the order of the group. The base case for induction, namely the case of a group of prime order, is clear.

For the induction step, suppose the result is true for all groups whose order is $p^d,d<r$. We want to show that the result is true for $p^r$. Let $G$ be a group of order $p^r,r\ge 1$. Then since any nontrivial group of prime power order has nontrivial center, $Z(G)$ is nontrivial, and thus $G/Z(G)$ has order $p^d$ with $d<r$. Thus $G/Z(G)$ is nilpotent, so has an upper central series, and pulling this back gives an upper central series for $G$.