p-group

A \(p\)-group can be defined in two equivalent ways: by its order, which is \(p^n\), or by its elements, since every element has order a power of \(p\). One direction comes from Lagrange (the order of each element divides \(p^n\)); the other is Cauchy's theorem, which guarantees an element of order \(q\) for each prime \(q\) dividing \(|G|\). Likewise, every subgroup and every quotient of a \(p\)-group is again a \(p\)-group.

The central property is that the center of a nontrivial \(p\)-group is nontrivial. Let \(G\) act on itself by conjugation: each orbit size divides \(p^n\), and the orbits of size \(1\) are the elements of the center; since \(p^n\) is a multiple of \(p\), so is \(|Z(G)|\), hence \(|Z(G)|\ge p\). More generally, if a \(p\)-group acts on a set \(X\), then \(|X|\equiv|X^G|\pmod p\), where \(X^G\) is the set of fixed points.

Guiding examples: \(\mathbb{Z}_{p^n}\), the groups \(\mathbb{Z}_p^k\) and the symmetry group of the square, of order \(8\), are \(p\)-groups. Groups of order \(p\) are cyclic, and those of order \(p^2\) are always abelian (cyclic or \(\mathbb{Z}_p\times\mathbb{Z}_p\)). At order \(p^3\) nonabelian groups already appear, so commutativity must not be assumed.

Typical mistake: thinking that a group whose order has several primes is a «\(p\)-group of some kind». A group of order \(12\) is not a \(p\)-group, even though it has \(2\)-subgroups and \(3\)-subgroups; those maximal subgroups are precisely the Sylow ones, the topic that follows.