- P-group
In
mathematics , given aprime number "p", a "p"-group is aperiodic group in which each element has a power of "p" as its order. That is, for each element "g" of the group, there exists anonnegative integer "n" such that "g" to the power "pn" is equal to theidentity 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, seeTarski 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 aproper subgroup "H" of a finite "p"-group "G" properly contains "H", because for anycounterexample 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 "i"th 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 "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 nontrivialouter automorphism group . Every automorphism of "G" induces an automorphism on "G"/Φ("G"), where Φ("G") is theFrattini subgroup of "G". The quotient G/Φ("G") is anelementary abelian group and itsautomorphism group is ageneral 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 "C"4 and theKlein group "V"4 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 "p"2 is abelian.The dihedral groups are both very similar to and very dissimilar from the
quaternion group s and thesemidihedral group s. Together the dihedral, semidihedral, and quaternion groups form the 2-groups ofmaximal class , that is those groups of order 2"n"+1 and nilpotency class "n".Iterated wreath products
The iterated
wreath product s 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 thesymmetric group Sym("p""n"). 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"=("p""n"−1)/("p"−1). It has nilpotency class "p""n"−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 "p""n". 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 2"n", but that requires a bit more setup. Let ζ denote a primitive "p"th root of unity in the complex numbers, and let Z [ζ] be the ring of
cyclotomic integer s generated by it, and let "P" be theprime ideal generated by 1−ζ. Let "G" be a cyclic group of order "p" generated by an element "z". Form thesemidirect 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 "p"-group 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 group s 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/"p"Z, then "U"1 is a Sylow "p"-subgroup of Aut("V") = GL("n", "p"), and the terms of itslower 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 harv|Burnside|1897, 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 harv|Leedham-Green|McKay|2002.
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 harv|Hall|1940.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 group s. The coclass conjectures were proven in the 1980s using techniques related toLie algebra s andpowerful p-group s harv|Leedham-Green|McKay|2002.Prevalence
In an asymptotic sense, almost all
finite group s are "p"-groups. Infact, almost all finite groups are 2-groups: the fraction ofisomorphism class es 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 harv|Besche|Eick|O'Brien|2002.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
normalizer s of non-identity "p"-subgroups harv|Glauberman|1971.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 calledextraspecial group s help describe the structure of groups as actingsymplectic vector space s.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.
See also
*
nilpotent group
*Prüfer rank
*Regular p-group References
*Citation | last1=Besche | first1=Hans Ulrich | last2=Eick | first2=Bettina | last3=O'Brien | first3=E. A. | title=A millennium project: constructing small groups | id=MathSciNet | id = 1935567 | year=2002 | journal=International Journal of Algebra and Computation | issn=0218-1967 | volume=12 | issue=5 | pages=623–644
*
* | year=1971 | chapter=Global and local properties of finite groups | pages=1–64
* | year=1940 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=182 | pages=130–141
* | year=2002 | volume=27
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