p-group
Algebraic structure → Group theory Group theory |
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In
Abelian p-groups are also called p-primary or simply primary.
A
Every finite p-group is nilpotent.
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.
Properties
Every p-group is
If p is prime and G is a group of order pk, then G has a normal subgroup of order pm for every 1 ≤ m ≤ k. This follows by induction, using
Non-trivial center
One of the first standard results using the
This forms the basis for many inductive methods in p-groups.
For instance, the
In another direction, every
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
Automorphisms
The
Examples
p-groups of the same order are not necessarily isomorphic; for example, the cyclic group C4 and the Klein four-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.[note 1]
The dihedral groups are both very similar to and very dissimilar from the
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 pk 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 pp+1 and nilpotency class p, but is not a regular p-group. Since groups of order pp 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 pp+1, but are not isomorphic.
Unitriangular matrix groups
The Sylow subgroups of
Classification
The groups of order pn 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 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.[3] For example, Marshall Hall Jr. and James K. Senior classified groups of order 2n 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 p-groups into families based on large quotient and subgroups.[5]
An entirely different method classifies finite p-groups by their
Every group of order p5 is metabelian.[7]
Up to p3
The trivial group is the only group of order one, and the cyclic group Cp is the only group of order p. There are exactly two groups of order p2, both abelian, namely Cp2 and Cp × Cp. 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 Cp3, Cp2 × Cp, and Cp × Cp × Cp. There are also two non-abelian groups.
For p ≠ 2, one is a semi-direct product of Cp × Cp with Cp, and the other is a semi-direct product of Cp2 with Cp. 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.
Prevalence
Among groups
The number of isomorphism classes of groups of order pn grows as , 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 where p does not divide m, then G has a subgroup P of order 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.
Application to structure of a group
p-groups are fundamental tools in understanding the structure of groups and in the
Local control
Much of the structure of a finite group is carried in the structure of its so-called local subgroups, the
The large
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.
See also
Footnotes
Notes
- ^ To prove that a group of order p2 is abelian, note that it is a p-group so has non-trivial center, so given a non-trivial 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)
- ^ (Leedham-Green & McKay 2002, p. 214)
- ^ (Hall Jr. & Senior 1964)
- ^ (Hall 1940)
- ^ (Leedham-Green & McKay 2002)
- ^ "Every group of order p5 is metabelian". Stack Exchange. 24 March 2012. Retrieved 7 January 2016.
- ^ (Sims 1965)
- .
- ^ (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, S2CID 31716675
- ISBN 9781440035456
- MR 0352241
- finite groups" (from the preface).
- S2CID 122817195
- MR 1918951
- Sims, Charles (1965), "Enumerating p-groups", Proc. London Math. Soc., Series 3, 15: 151–166, MR 0169921
Further reading
- ISBN 978-3-1102-0418-6
- Berkovich, Yakov; ISBN 978-3-1102-0419-3
- 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, ISBN 978-3-1102-0717-0