Throughout this chapter, p denotes a prime number. We prove here some elementary properties of finite p-groups including the Burnside Basis Theorem. Then we prove a theorem of P. Hall on the orders of the lower central factors of a normal subgroup. Many other properties of finite p-groups will be proved later, some in Chapter 6 using the associated Lie rings, some in Chapter 10 using the Mal'cev–Lazard correspondence, some in Chapter 11 on powerful p-groups. The main results of the book in Chapters 8, 12, 13 and 14 are also about finite p-groups.

We shall freely use the fact that the homomorphic images of commutator subgroups are commutator subgroups of the images (1.14), the same being true for verbal subgroups, like Gn = 〈gn | g ∈ G〉, by Lemma 1.47.

Basic properties

By the definition from § 1.1, a group is a p-group if the orders of all of its elements are powers of p. By Lagrange's Theorem, any group of order pn, n ∈ N, is a finite p-group. The converse is also true by the Sylow Theorems. Thus, we can safely redefine finite p-groups as groups of order pn, n ∈ N. By Lagrange's Theorem, all factor-groups and all subgroups of a finite p-group are again finite p-groups. Note that every group of order p is cyclic, since every non-trivial element generates a subgroup of order that divides p and hence equals p.