Non-trivial center

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.^[1]

This forms the basis for many inductive methods in p-groups.

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. As a corollary, every finite p-group is nilpotent.

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. The preimage in G of the center of G/Z is called the second center and these groups begin the upper central series. Generalizing the earlier comments about the socle, a finite p-group with order pⁿ contains normal subgroups of order pⁱ with 0 ≤ i ≤ n, and any normal subgroup of order pⁱ is contained in the ith center Z_i. If a normal subgroup is not contained in Z_i, then its intersection with Z_i+1 has size at least pⁱ⁺¹.

Automorphisms

The automorphism groups of p-groups are well studied. Just as every finite p-group has a non-trivial center so that the inner automorphism group is a proper quotient of the group, every finite p-group has a non-trivial outer automorphism group. Every automorphism of G induces an automorphism on G/Φ(G), where Φ(G) is the Frattini subgroup of G. The quotient G/Φ(G) is an elementary abelian group and its automorphism group is a general linear group, so very well understood. The map from the automorphism group of G into this general linear group has been studied by Burnside, who showed that the kernel of this map is a p-group.

Iterated wreath products

The iterated wreath products of cyclic groups of order p are very important examples of p-groups. Denote the cyclic group of order p as W(1), and the wreath product of W(n) with W(1) as W(n + 1). Then W(n) is the Sylow p-subgroup of the symmetric group Sym(pⁿ). Maximal p-subgroups of the general linear group GL(n,Q) are direct products of various W(n). It has order p^k where k = (pⁿ − 1)/(p − 1). It has nilpotency class pⁿ⁻¹, and its lower central series, upper central series, lower exponent-p central series, and upper exponent-p central series are equal. It is generated by its elements of order p, but its exponent is pⁿ. The second such group, W(2), is also a p-group of maximal class, since it has order p^p+1 and nilpotency class p, but is not a regular p-group. Since groups of order p^p are always regular groups, it is also a minimal such example.

Generalized dihedral groups

When p = 2 and n = 2, W(n) is the dihedral group of order 8, so in some sense W(n) provides an analogue for the dihedral group for all primes p when n = 2. However, for higher n the analogy becomes strained. 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. Let G be a cyclic group of order p generated by an element z. Form the semidirect product E(p) of Z[ζ] and G where z acts as multiplication by ζ. The powers Pⁿ are normal subgroups of E(p), and the example groups are E(p,n) = E(p)/Pⁿ. E(p,n) has order pⁿ⁺¹ and nilpotency class n, so is a p-group of maximal class. When p = 2, E(2,n) is the dihedral group of order 2ⁿ. When p is odd, both W(2) and E(p,p) are irregular groups of maximal class and order p^p+1, but are not isomorphic.

Unitriangular matrix groups

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, ..., e_n } and define V_i to be the vector space generated by { e_i, e_i+1, ..., e_n } for 1 ≤ i ≤ n, and define V_i = 0 when i > n. For each 1 ≤ m ≤ n, the set of invertible linear transformations of V which take each V_i to V_i+m form a subgroup of Aut(V) denoted U_m. 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 U_m. In terms of matrices, U_m are those upper triangular matrices with 1s one the diagonal and 0s on the first m−1 superdiagonals. The group U₁ has order p^n·(n−1)/2, nilpotency class n, and exponent p^k where k is the least integer at least as large as the base p logarithm of n.

Up to p³

The trivial group is the only group of order one, and the cyclic group C_p is the only group of order p. There are exactly two groups of order p², both abelian, namely C_p² and C_p × C_p. For example, the cyclic group C₄ and the Klein four-group V₄ which is C₂ × C₂ are both 2-groups of order 4.

There are three abelian groups of order p³, namely C_p³, C_p² × C_p, and C_p × C_p × C_p. There are also two non-abelian groups.

For p ≠ 2, one is a semi-direct product of C_p × C_p with C_p, and the other is a semi-direct product of C_p² with C_p. The first one can be described in other terms as group UT(3,p) of unitriangular matrices over finite field with p elements, also called the Heisenberg group mod p.

For p = 2, both the semi-direct products mentioned above are isomorphic to the dihedral group Dih₄ of order 8. The other non-abelian group of order 8 is the quaternion group Q₈.

Among groups

The number of isomorphism classes of groups of order pⁿ grows as ${\displaystyle p^{{\frac {2}{27}}n^{3}+O(n^{8/3})}}$, and these are dominated by the classes that are two-step nilpotent.^[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. For instance, of the 49 910 529 484 different groups of order at most 2000, 49487367289, or just over 99%, are 2-groups of order 1024.^[9]

Within a group

Every finite group whose order is divisible by p contains a subgroup which is a non-trivial p-group, namely a cyclic group of order p generated by an element of order p obtained from Cauchy's theorem. In fact, it contains a p-group of maximal possible order: if ${\displaystyle |G|=n=p^{k}m}$ where p does not divide m, then G has a subgroup P of order ${\displaystyle p^{k},}$ called a Sylow p-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any p-subgroup of G is contained in a Sylow p-subgroup. This and other properties are proved in the Sylow theorems.

Local control

Much of the structure of a finite group is carried in the structure of its so-called local subgroups, the normalizers of non-identity p-subgroups.^[10]

The large elementary abelian subgroups of a finite group exert control over the group that was used in the proof of the Feit–Thompson theorem. Certain central extensions of elementary abelian groups called extraspecial groups help describe the structure of groups as acting on symplectic vector spaces.

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.

Notes

Citations