A finite group is a p-group if and only if its order (the number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime power pn that divides the order of G. Every finite p-group is nilpotent.
Every p-group is periodic since by definition every element has finite order.
If p is prime and G is a group of order pk, then G has a normal subgroup of order pm for every 1 ≤ m ≤ k. This follows by induction, using Cauchy's theorem and the Correspondence Theorem for groups.
A proof sketch is as follows: because the center Z of G is non-trivial (see below), according to Cauchy's theorem Z has a subgroup H of order p. Being central in G, H is necessarily normal in G. We may now apply the inductive hypothesis to G/H, and the result follows from the Correspondence Theorem.
One of the first standard results using the class equation is that the center of a non-trivial finite p-group cannot be the trivial subgroup.
For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H/Z whose normalizer in G/Z is N/Z = H/Z, creating an infinite descent.
In another direction, every normal subgroup N of a finite p-group intersects the center non-trivially as may be proved by considering the elements of N which are fixed when G acts on N by conjugation.
Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite p-group is central and has order p. Indeed, the socle of a finite p-group is the subgroup of the center consisting of the central elements of order p. If G is a p-group, then so is G/Z, and so it too has a non-trivial center.
Nor need a p-group be abelian; the dihedral group Dih4 of order 8 is a non-abelian 2-group.
Maximal p-subgroups of the general linear group GL(n,Q) are direct products of various W(n).
There is a different family of examples that more closely mimics the dihedral groups of order 2n, but that requires a bit more setup.
Let ζ denote a primitive pth root of unity in the complex numbers, let Z[ζ] be the ring of cyclotomic integers generated by it, and let P be the prime ideal generated by 1−ζ.
When p is odd, both W(2) and E(p,p) are irregular groups of maximal class and order pp+1, but are not isomorphic.
The Sylow subgroups of general linear groups are another fundamental family of examples.
Let V be a vector space of dimension n with basis { e1, e2, ..., en } and define Vi to be the vector space generated by { ei, ei+1, ..., en } for 1 ≤ i ≤ n, and define Vi = 0 when i > n. For each 1 ≤ m ≤ n, the set of invertible linear transformations of V which take each Vi to Vi+m form a subgroup of Aut(V) denoted Um.
If V is a vector space over Z/pZ, then U1 is a Sylow p-subgroup of Aut(V) = GL(n, p), and the terms of its lower central series are just the Um.
[3] For example, Marshall Hall Jr. and James K. Senior classified groups of order 2n for n ≤ 6 in 1964.
[4] Rather than classify the groups by order, Philip Hall proposed using a notion of isoclinism of groups which gathered finite p-groups into families based on large quotient and subgroups.
The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful p-groups.
[6] The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994.
The number of isomorphism classes of groups of order pn grows as
[8] Because of this rapid growth, there is a folklore conjecture asserting that almost all finite groups are 2-groups: the fraction of isomorphism classes of 2-groups among isomorphism classes of groups of order at most n is thought to tend to 1 as n tends to infinity.
Richard Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and John Walter, Daniel Gorenstein, Helmut Bender, Michio Suzuki, George Glauberman, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.