pgroup
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

In mathematical group theory, given a prime number p, a pgroup is a group in which each element has a power of p as its order. That is, for each element g of a pgroup, there exists a nonnegative integer n such that the product of p^{n} copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
Abelian pgroups are also called pprimary or simply primary.
A finite group is a pgroup 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 for every prime power p^{n} that divides the order of G the existence of a subgroup of G of order p^{n}.
The remainder of this article deals with finite pgroups. For an example of an infinite abelian pgroup, see Prüfer group, and for an example of an infinite simple pgroup, see Tarski monster group.
Contents
Properties
Every pgroup is periodic since by definition every element has finite order.
If p is prime and G is a group of order p^{k}, then G has a normal subgroup of order p^{m} 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: by Cauchy's Theorem, G has a subgroup H of order p. If the normalizer of H in G is G itself, then we may apply the inductive hypothesis to G/H, and the result follows from the Correspondence Theorem. Otherwise, we apply the same logic to G/N_{G}(H), where N_{G}(H) is the normalizer of H in G.
Nontrivial center
One of the first standard results using the class equation is that the center of a nontrivial finite pgroup cannot be the trivial subgroup.^{[1]}
This forms the basis for many inductive methods in pgroups.
For instance, the normalizer N of a proper subgroup H of a finite pgroup 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 pgroup is nilpotent.
In another direction, every normal subgroup of a finite pgroup intersects the center nontrivially 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 pgroup is central and has order p. Indeed, the socle of a finite pgroup is the subgroup of the center consisting of the central elements of order p.
If G is a pgroup, 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 pgroup with order p^{n} contains normal subgroups of order p^{i} with 0 ≤ i ≤ n, and any normal subgroup of order p^{i} 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 p^{i+1}.
Automorphisms
The automorphism groups of pgroups are well studied. Just as every finite pgroup has a nontrivial center so that the inner automorphism group is a proper quotient of the group, every finite pgroup 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 pgroup.
Examples
pgroups of the same order are not necessarily isomorphic; for example, the cyclic group C_{4} and the Klein fourgroup V_{4} are both 2groups of order 4, but they are not isomorphic.
Nor need a pgroup be abelian; the dihedral group Dih_{4} of order 8 is a nonabelian 2group. However, every group of order p^{2} is abelian.^{[note 1]}
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 2groups of maximal class, that is those groups of order 2^{n+1} and nilpotency class n.
Iterated wreath products
The iterated wreath products of cyclic groups of order p are very important examples of pgroups. 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 psubgroup of the symmetric group Sym(p^{n}). Maximal psubgroups of the general linear group GL(n,Q) are direct products of various W(n). It has order p^{k} where k = (p^{n} − 1)/(p − 1). It has nilpotency class p^{n−1}, and its lower central series, upper central series, lower exponentp central series, and upper exponentp central series are equal. It is generated by its elements of order p, but its exponent is p^{n}. The second such group, W(2), is also a pgroup of maximal class, since it has order p^{p+1} and nilpotency class p, but is not a regular pgroup. 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 2^{n}, 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 P^{n} are normal subgroups of E(p), and the example groups are E(p,n) = E(p)/P^{n}. E(p,n) has order p^{n+1} and nilpotency class n, so is a pgroup of maximal class. When p = 2, E(2,n) is the dihedral group of order 2^{n}. 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 { e_{1}, e_{2}, …, 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 U_{1} is a Sylow psubgroup 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 U_{1} 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 p^{n} for 0 ≤ n ≤ 4 were classified early in the history of group theory,^{[2]} and modern work has extended these classifications to groups whose order divides p^{7}, 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.^{[3]} For example, Marshall Hall Jr. and James K. Senior classified groups of order 2^{n} 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 pgroups into families based on large quotient and subgroups.^{[5]}
An entirely different method classifies finite pgroups by their coclass, that is, the difference between their composition length and their nilpotency class. The socalled coclass conjectures described the set of all finite pgroups of fixed coclass as perturbations of finitely many prop groups. The coclass conjectures were proven in the 1980s using techniques related to Lie algebras and powerful pgroups.^{[6]} The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. LeedhamGreen, both in 1994. They admit a classification of finite pgroups in directed coclass graphs consisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.
Every group of order p^{5} is metabelian.^{[7]}
Up to p^{3}
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 p^{2}, both abelian, namely C_{p2} and C_{p} × C_{p}. For example, the cyclic group C_{4} and the Klein fourgroup V_{4} which is C_{2} × C_{2} are both 2groups of order 4.
There are three abelian groups of order p^{3}, namely C_{p3}, C_{p2}×C_{p}, and C_{p}×C_{p}×C_{p}. There are also two nonabelian groups.
For p ≠ 2, one is a semidirect product of C_{p}×C_{p} with C_{p}, and the other is a semidirect 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 semidirect products mentioned above are isomorphic to the dihedral group Dih_{4} of order 8. The other nonabelian group of order 8 is the quaternion group Q_{8}.
Prevalence
Among groups
The number of isomorphism classes of groups of order p^{n} grows as , and these are dominated by the classes that are twostep nilpotent.^{[8]} Because of this rapid growth, there is a folklore conjecture asserting that almost all finite groups are 2groups: the fraction of isomorphism classes of 2groups 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, 49 487 365 422, or just over 99%, are 2groups of order 1024.^{[9]}
Within a group
Every finite group whose order is divisible by p contains a subgroup which is a nontrivial pgroup, namely a cyclic group of order p generated by an element of order p obtained from Cauchy's theorem. In fact, it contains a pgroup of maximal possible order: if where p does not divide m, then G has a subgroup P of order called a Sylow psubgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any psubgroup of G is contained in a Sylow psubgroup. This and other properties are proved in the Sylow theorems.
Application to structure of a group
pgroups are fundamental tools in understanding the structure of groups and in the classification of finite simple groups. pgroups arise both as subgroups and as quotient groups. As subgroups, for a given prime p one has the Sylow psubgroups P (largest psubgroup not unique but all conjugate) and the pcore (the unique largest normal psubgroup), and various others. As quotients, the largest pgroup quotient is the quotient of G by the presidual subgroup These groups are related (for different primes), possess important properties such as the focal subgroup theorem, and allow one to determine many aspects of the structure of the group.
Local control
Much of the structure of a finite group is carried in the structure of its socalled local subgroups, the normalizers of nonidentity psubgroups.^{[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.
Brauer classified all groups whose Sylow 2subgroups 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 2subgroups were abelian, dihedral, semidihedral, or quaternion.
See also
Footnotes
Notes
 ^ To prove that a group of order p^{2} is abelian, note that it is a pgroup so has nontrivial center, so given a nontrivial element of the center g, this either generates the group (so G is cyclic, hence abelian: ), or it generates a subgroup of order p, so g and some element h not in its orbit generate G, (since the subgroup they generate must have order ) but they commute since g is central, so the group is abelian, and in fact
Citations
 ^ proof
 ^ (Burnside 1897)
 ^ (LeedhamGreen & McKay 2002, p. 214)
 ^ (Hall, Jr. & Senior 1964)
 ^ (Hall 1940)
 ^ (LeedhamGreen & McKay 2002)
 ^ "Every group of order p5 is metabelian". Stack Exchange. 24 March 2012. Retrieved 7 January 2016.
 ^ (Sims 1965)
 ^ (Besche, Eick & O'Brien 2002)
 ^ (Glauberman 1971)
References
 Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2002), "A millennium project: constructing small groups", International Journal of Algebra and Computation, 12 (5): 623–644, doi:10.1142/S0218196702001115, MR 1935567
 Burnside, William (1897), Theory of groups of finite order, Cambridge University Press
 Glauberman, George (1971), "Global and local properties of finite groups", Finite simple groups (Proc. Instructional Conf., Oxford, 1969), Boston, MA: Academic Press, pp. 1–64, MR 0352241
 Hall, Jr., Marshall; Senior, James K. (1964), The Groups of Order 2^{n} (n ≤ 6), London: Macmillan, LCCN 64016861, MR 0168631 — An exhaustive catalog of the 340 nonabelian groups of order dividing 64 with detailed tables of defining relations, constants, and lattice presentations of each group in the notation the text defines. "Of enduring value to those interested in finite groups" (from the preface).
 Hall, Philip (1940), "The classification of primepower groups", Journal für die reine und angewandte Mathematik, 182 (182): 130–141, doi:10.1515/crll.1940.182.130, ISSN 00754102, MR 0003389
 LeedhamGreen, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series, 27, Oxford University Press, ISBN 9780198535485, MR 1918951
 Sims, Charles (1965), "Enumerating pgroups", Proc. London Math. Soc., Series 3, 15: 151–166, doi:10.1112/plms/s315.1.151, MR 0169921
Further reading
 Berkovich, Yakov (2008), Groups of Prime Power Order, de Gruyter Expositions in Mathematics 46, Volume 1, Berlin: Walter de Gruyter GmbH, ISBN 9783110204186
 Berkovich, Yakov; Janko, Zvonimir (2008), Groups of Prime Power Order, de Gruyter Expositions in Mathematics 47, Volume 2, Berlin: Walter de Gruyter GmbH, ISBN 9783110204193
 Berkovich, Yakov; Janko, Zvonimir (20110616), Groups of Prime Power Order, de Gruyter Expositions in Mathematics 56, Volume 3, Berlin: Walter de Gruyter GmbH, ISBN 9783110207170
External links
 Rowland, Todd and Weisstein, Eric W. "pGroup". MathWorld.