p-group

Not to be confused withn-group (category theory).

Inmathematics,specificallygroup theory,given aprime numberp,ap-groupis agroupin which theorderof every element is apowerofp.That is, for each elementgof ap-groupG,there exists anonnegative integernsuch that the product ofpⁿcopies ofg,and not fewer, is equal to theidentity element.The orders of different elements may be different powers ofp.

Abelianp-groups are also calledp-primaryor simplyprimary.

Afinite groupis ap-group if and only if itsorder(the number of its elements) is a power ofp.Given a finite groupG,theSylow theoremsguarantee the existence of asubgroupofGof orderpⁿfor everyprime powerpⁿthat divides the order ofG.

Every finitep-group isnilpotent.

The remainder of this article deals with finitep-groups. For an example of an infinite abelianp-group, seePrüfer group,and for an example of an infinitesimplep-group, seeTarski monster group.

Properties

Everyp-group isperiodicsince by definition every element hasfinite order.

Ifpis prime andGis a group of orderp^k,thenGhas a normal subgroup of orderp^mfor every 1 ≤m≤k.This follows by induction, usingCauchy's theoremand theCorrespondence Theoremfor groups. A proof sketch is as follows: because thecenterZofGisnon-trivial(see below), according toCauchy's theoremZhas a subgroupHof orderp.Being central inG,His necessarily normal inG.We may now apply the inductive hypothesis toG/H,and the result follows from the Correspondence Theorem.

Non-trivial center

One of the first standard results using theclass equationis that thecenterof a non-trivial finitep-group cannot be the trivial subgroup.

This forms the basis for many inductive methods inp-groups.

For instance, thenormalizerNof aproper subgroupHof a finitep-groupGproperly containsH,because for anycounterexamplewithH=N,the centerZis contained inN,and so also inH,but then there is a smaller exampleH/Zwhose normalizer inG/ZisN/Z=H/Z,creating an infinite descent. As a corollary, every finitep-group isnilpotent.

In another direction, everynormal subgroupNof a finitep-group intersects the center non-trivially as may be proved by considering the elements ofNwhich are fixed whenGacts onNby conjugation. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finitep-group is central and has orderp.Indeed, thesocleof a finitep-group is the subgroup of the center consisting of the central elements of orderp.

IfGis ap-group, then so isG/Z,and so it too has a non-trivial center. The preimage inGof the center ofG/Zis called thesecond centerand these groups begin theupper central series.Generalizing the earlier comments about the socle, a finitep-group with orderpⁿcontains normal subgroups of orderpⁱwith 0 ≤i≤n,and any normal subgroup of orderpⁱis contained in theith centerZ_i.If a normal subgroup is not contained inZ_i,then its intersection withZ_i+1has size at leastpⁱ⁺¹.

Automorphisms

Theautomorphismgroups ofp-groups are well studied. Just as every finitep-group has a non-trivial center so that theinner automorphism groupis a proper quotient of the group, every finitep-group has a non-trivialouter automorphism group.Every automorphism ofGinduces an automorphism onG/Φ(G), where Φ(G) is theFrattini subgroupofG.The quotient G/Φ(G) is anelementary abelian groupand itsautomorphism groupis ageneral linear group,so very well understood. The map from the automorphism group ofGinto this general linear group has been studied byBurnside,who showed that the kernel of this map is ap-group.

Examples

p-groups of the same order are not necessarilyisomorphic;for example, thecyclic groupC₄and theKlein four-groupV₄are both 2-groups of order 4, but they are not isomorphic.

Nor need ap-group beabelian;thedihedral groupDih₄of order 8 is a non-abelian 2-group. However, every group of orderp²is abelian.

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

Iterated wreath products

The iteratedwreath productsof cyclic groups of orderpare very important examples ofp-groups. Denote the cyclic group of orderpasW(1), and the wreath product ofW(n) withW(1) asW(n+ 1). ThenW(n) is the Sylowp-subgroup of thesymmetric groupSym(pⁿ). Maximalp-subgroups of the general linear group GL(n,Q) are direct products of variousW(n). It has orderp^kwherek= (pⁿ− 1)/(p− 1). It has nilpotency classpⁿ⁻¹,and its lower central series, upper central series, lower exponent-pcentral series, and upper exponent-pcentral series are equal. It is generated by its elements of orderp,but its exponent ispⁿ.The second such group,W(2), is also ap-group of maximal class, since it has orderp^p+1and nilpotency classp,but is not aregularp-group.Since groups of orderp^pare always regular groups, it is also a minimal such example.

Generalized dihedral groups

Whenp= 2 andn= 2,W(n) is the dihedral group of order 8, so in some senseW(n) provides an analogue for the dihedral group for all primespwhenn= 2. However, for highernthe 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 primitivepth root of unity in the complex numbers, letZ[ζ] be the ring ofcyclotomic integersgenerated by it, and letPbe theprime idealgenerated by 1−ζ. LetGbe a cyclic group of orderpgenerated by an elementz.Form thesemidirect productE(p) ofZ[ζ] andGwherezacts as multiplication by ζ. The powersPⁿare normal subgroups ofE(p), and the example groups areE(p,n) =E(p)/Pⁿ.E(p,n) has orderpⁿ⁺¹and nilpotency classn,so is ap-group of maximal class. Whenp= 2,E(2,n) is the dihedral group of order 2ⁿ.Whenpis odd, bothW(2) andE(p,p) are irregular groups of maximal class and orderp^p+1,but are not isomorphic.

Unitriangular matrix groups

The Sylow subgroups ofgeneral linear groupsare another fundamental family of examples. LetVbe a vector space of dimensionnwith basis {e₁,e₂,...,e_n} and defineV_ito be the vector space generated by {e_i,e_i+1,...,e_n} for 1 ≤i≤n,and defineV_i= 0 wheni>n.For each 1 ≤m≤n,the set of invertible linear transformations ofVwhich take eachV_itoV_i+mform a subgroup of Aut(V) denotedU_m.IfVis a vector space overZ/pZ,thenU₁is a Sylowp-subgroup of Aut(V) = GL(n,p), and the terms of itslower central seriesare just theU_m.In terms of matrices,U_mare those upper triangular matrices with 1s one the diagonal and 0s on the firstm−1 superdiagonals. The groupU₁has orderp^n·(n−1)/2,nilpotency classn,and exponentp^kwherekis the least integer at least as large as the baseplogarithmofn.

Classification

The groups of orderpⁿfor 0 ≤n≤ 4 were classified early in the history of group theory,and modern work has extended these classifications to groups whose order dividesp⁷,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.For example,Marshall Hall Jr.and James K. Senior classified groups of order 2ⁿforn≤ 6 in 1964.

Rather than classify the groups by order,Philip Hallproposed using a notion ofisoclinism of groupswhich gathered finitep-groups into families based on large quotient and subgroups.

An entirely different method classifies finitep-groups by theircoclass,that is, the difference between theircomposition lengthand theirnilpotency class.The so-calledcoclass conjecturesdescribed the set of all finitep-groups of fixed coclass as perturbations of finitely manypro-p groups.The coclass conjectures were proven in the 1980s using techniques related toLie algebrasandpowerful p-groups.The final proofs of thecoclass theoremsare due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finitep-groups indirected coclass graphsconsisting of only finitely many coclass trees whose (infinitely many) members are characterized by finitely many parametrized presentations.

Every group of orderp⁵ismetabelian.

Up top³

The trivial group is the only group of order one, and the cyclic group C_pis the only group of orderp.There are exactly two groups of orderp²,both abelian, namely C_p²and C_p× C_p.For example, the cyclic group C₄and theKlein four-groupV₄which is C₂× C₂are both 2-groups of order 4.

There are three abelian groups of orderp³,namely C_p³,C_p²× C_p,and C_p× C_p× C_p.There are also two non-abelian groups.

Forp≠ 2, one is a semi-direct product of C_p× C_pwith C_p,and the other is a semi-direct product of C_p²with C_p.The first one can be described in other terms as group UT(3,p) of unitriangular matrices over finite field withpelements, also called theHeisenberg group modp.

Forp= 2, both the semi-direct products mentioned above are isomorphic to thedihedral groupDih₄of order 8. The other non-abelian group of order 8 is thequaternion groupQ₈.

Prevalence

Among groups

The number of isomorphism classes of groups of orderpⁿgrows as$p^{{\frac {2}{27}}n^{3}+O(n^{8/3})}$,and these are dominated by the classes that are two-step nilpotent.Because of this rapid growth, there is afolkloreconjecture asserting that almost allfinite groupsare 2-groups: the fraction ofisomorphism classesof 2-groups among isomorphism classes of groups of order at mostnis thought to tend to 1 asntends 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.

Within a group

Every finite group whose order is divisible bypcontains a subgroup which is a non-trivialp-group, namely a cyclic group of orderpgenerated by an element of orderpobtained fromCauchy's theorem.In fact, it contains ap-group of maximal possible order: if$|G|=n=p^{k}m$wherepdoes not dividem,thenGhas a subgroupPof order$p^{k},$called a Sylowp-subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and anyp-subgroup ofGis contained in a Sylowp-subgroup. This and other properties are proved in theSylow theorems.

Application to structure of a group

p-groups are fundamental tools in understanding the structure of groups and in theclassification of finite simple groups.p-groups arise both as subgroups and as quotient groups. As subgroups, for a given primepone has the Sylowp-subgroupsP(largestp-subgroup not unique but all conjugate) and thep-core$O_{p}(G)$(the unique largestnormalp-subgroup), and various others. As quotients, the largestp-group quotient is the quotient ofGby thep-residual subgroup$O^{p}(G).$These groups are related (for different primes), possess important properties such as thefocal 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 so-calledlocal subgroups,thenormalizersof non-identityp-subgroups.

The largeelementary abelian subgroupsof a finite group exert control over the group that was used in the proof of theFeit–Thompson theorem.Certaincentral extensionsof elementary abelian groups calledextraspecial groupshelp describe the structure of groups as acting onsymplectic vector spaces.

Richard Brauerclassified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, andJohn 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.

See also

• Elementary group
• Prüfer rank
• Regular p-group

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,MR1935567,S2CID31716675
• Burnside, William(1897),Theory of groups of finite order,Cambridge University Press,ISBN9781440035456
• 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,MR0352241
• Hall Jr., Marshall;Senior, James K. (1964),The Groups of Order 2ⁿ(n≤ 6),London: Macmillan,LCCN64016861,MR0168631— An exhaustive catalog of the 340 non-abelian groups of order dividing 64 with detailed tables of defining relations, constants, andlatticepresentations of each group in the notation the text defines. "Of enduring value to those interested infinite groups"(from the preface).
• Hall, Philip(1940),"The classification of prime-power groups",Journal für die reine und angewandte Mathematik,1940(182): 130–141,doi:10.1515/crll.1940.182.130,ISSN0075-4102,MR0003389,S2CID122817195
• Leedham-Green, C. R.;McKay, Susan (2002),The structure of groups of prime power order,London Mathematical Society Monographs. New Series, vol. 27,Oxford University Press,ISBN978-0-19-853548-5,MR1918951
• Sims, Charles (1965), "Enumerating p-groups",Proc. London Math. Soc.,Series 3,15:151–166,doi:10.1112/plms/s3-15.1.151,MR0169921

Further reading

• Berkovich, Yakov(2008),Groups of Prime Power Order,de Gruyter Expositions in Mathematics 46, vol. 1, Berlin: Walter de Gruyter GmbH,ISBN978-3-1102-0418-6
• Berkovich, Yakov;Janko, Zvonimir(2008),Groups of Prime Power Order,de Gruyter Expositions in Mathematics 47, vol. 2, Berlin: Walter de Gruyter GmbH,ISBN978-3-1102-0419-3
• Berkovich, Yakov; Janko, Zvonimir (2011-06-16),Groups of Prime Power Order,de Gruyter Expositions in Mathematics 56, vol. 3, Berlin: Walter de Gruyter GmbH,ISBN978-3-1102-0717-0

External links

• Rowland, Todd&Weisstein, Eric W."p-Group".MathWorld.