imido-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pⁿ is equal to the identity element. Such groups are also called primary.

The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.

A finite group is a p-group if and only if its order (the number of its elements) is a power of p.

Properties

Quite a lot is known about the structure of finite p-groups.

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 (proof).

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 of a finite p-group intersects the center nontrivially. In particular, every minimal normal subgroup of a finite p-group is of order p and contained in the center. 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 nontrivial 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 nontrivial center so that the inner automorphism group is a proper quotient of the group, every finite p-group has a nontrivial 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.

Examples

p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C₄ and the Klein group V₄ are both 2-groups of order 4, but they are not isomorphic.

Nor need a p-group be abelian; the dihedral group Dih₄ of order 8 is a non-abelian 2-group. However, every group of order p² is abelian.

The dihedral groups are both very similar to and very dissimilar from the quaternion groups and the semidihedral groups. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of maximal class, that is those groups of order 2ⁿ⁺¹ and nilpotency class n.

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, and 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.

Classification

The groups of order pⁿ for 0 ≤ n ≤ 4 were classified early in the history of group theory , and modern work has extended these classifications to groups whose order divides p⁷, though the sheer number of families of such groups grows so quickly that further classifications along these lines are judged difficult for the human mind to comprehend .

Rather than classify the groups by order, Philip Hall proposed using a notion of isoclinism which gathered finite p-groups into families based on large quotient and subgroups .

An entirely different method classifies finite p-groups by their coclass, that is, the difference between their composition length and their nilpotency class. The so-called coclass conjectures described the set of all finite p-groups of fixed coclass as perturbations of finitely many pro-p groups. The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful p-groups .

Prevalence

In an asymptotic sense, almost all finite groups are p-groups. In fact, 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 tends to 1 as n tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000, 49 487 365 422, or just over 99%, are 2-groups of order 1024 .

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, or a larger p-subgroup obtained from Sylow's theorem.

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 .

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 symplectic vector spaces.

Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and Walter, Gorenstein, Bender, Suzuki, Glauberman, and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.