In
mathematics, specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, given a
prime number ''p'', a ''p''-group is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
in which the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of every element is a
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a
nonnegative integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
''n'' such that the product of ''p
n'' copies of ''g'', and not fewer, is equal to the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
. The orders of different elements may be different powers of ''p''.
Abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
''p''-groups are also called ''p''-primary or simply primary.
A
finite group is a ''p''-group if and only if its
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
(the number of its elements) is a power of ''p''. Given a finite group ''G'', the
Sylow theorems guarantee the existence of a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of ''G'' of order ''p
n'' for every
prime power ''p
n'' that divides the order of ''G''.
Every finite ''p''-group is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
.
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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
''p''-group, see
Tarski monster group
In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group ''G'', such that every proper subgroup ''H'' of ''G'', other than the identity subgroup, is a cyclic group of order a fixe ...
.
Properties
Every ''p''-group is
periodic since by definition every element has
finite order
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subg ...
.
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: because the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
''Z'' of ''G'' is
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to ...
(see below), according to
Cauchy's theorem ''Z'' has a subgroup ''H'' of order ''p''. Being central in ''G'', ''H'' is necessarily normal in ''G''. We may now apply the inductive hypothesis to ''G/H'', and the result follows from the Correspondence Theorem.
Non-trivial center
One of the first standard results using the
class equation
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
is that the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of a non-trivial finite ''p''-group cannot be the trivial subgroup.
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
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
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 mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
.
In another direction, every
normal subgroup ''N'' of a finite ''p''-group intersects the center non-trivially 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 ''p''-group is central and has order ''p''. 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 non-trivial 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 ''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 ''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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
groups of ''p''-groups are well studied. Just as every finite ''p''-group has a non-trivial center so that the
inner automorphism group is a proper quotient of the group, every finite ''p''-group has a non-trivial
outer automorphism group. Every automorphism of ''G'' induces an automorphism on ''G''/Φ(''G''), where Φ(''G'') is the
Frattini subgroup
In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is def ...
of ''G''. The quotient G/Φ(''G'') is an
elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
and its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
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 ''C''
4 and the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
''V''
4 are both 2-groups of order 4, but they are not isomorphic.
Nor need a ''p''-group be
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
; the
dihedral group Dih
4 of order 8 is a non-abelian 2-group. However, every group of order ''p''
2 is abelian.
[To prove that a group of order ''p''2 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 ]
The dihedral groups are both very similar to and very dissimilar from the
quaternion groups and the
semidihedral group
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer ''n'' greater than or equal to 4, there are exactly four isomorphism classes of non ...
s. Together the dihedral, semidihedral, and quaternion groups form the 2-groups 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 ''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(''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, let Z
�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
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in ...
''E''(''p'') of Z
�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 groups are another fundamental family of examples. Let ''V'' be a vector space of dimension ''n'' with basis and define ''V''
''i'' to be the vector space generated by 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 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, 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. For example,
Marshall Hall Jr. and James K. Senior classified groups of order 2
''n'' for ''n'' ≤ 6 in 1964.
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.
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
In mathematics, a pro-''p'' group (for some prime number ''p'') is a profinite group G such that for any open normal subgroup N\triangleleft G the quotient group G/N is a ''p''-group. Note that, as profinite groups are compact, the open subgroups ...
s. The coclass conjectures were proven in the 1980s using techniques related to
Lie algebras and
powerful p-groups. The final proofs of the coclass theorems are due to A. Shalev and independently to C. R. Leedham-Green, both in 1994. They admit a classification of finite ''p''-groups 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.
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
''p''2 and C
''p'' × C
''p''. For example, the cyclic group C
4 and the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
''V''
4 which is C
2 × C
2 are both 2-groups of order 4.
There are three abelian groups of order ''p''
3, namely C
''p''3, C
''p''2 × C
''p'', and C
''p'' × C
''p'' × C
''p''. There are also two non-abelian groups.
For ''p'' ≠2, one is a semi-direct product of C
''p'' × C
''p'' with C
''p'', and the other is a semi-direct product of C
''p''2 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 semi-direct products mentioned above are isomorphic to the
dihedral group Dih
4 of order 8. The other non-abelian 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 two-step nilpotent. Because of this rapid growth, there is a
folklore conjecture asserting that almost all
finite groups are 2-groups: the fraction of
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the str ...
es 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, 49 487 365 422, or just over 99%, are 2-groups of order 1024.
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
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else ...
. ''p''-groups arise both as subgroups and as quotient groups. As subgroups, for a given prime ''p'' one has the Sylow ''p''-subgroups ''P'' (largest ''p''-subgroup not unique but all conjugate) and the
''p''-core (the unique largest ''normal'' ''p''-subgroup), and various others. As quotients, the largest ''p''-group quotient is the quotient of ''G'' by the
''p''-residual 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 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 In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by .
History
conjectured that every nonabelian finite simple group has even order. suggested using t ...
. Certain
central extensions of elementary abelian groups called
extraspecial group In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime ''p'' and positive integer ''n'' there are exactly two (up to isomorphism) extraspeci ...
s help describe the structure of groups as acting on
symplectic vector spaces.
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
Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissert ...
,
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
Footnotes
Notes
Citations
References
*
*
*
* — An exhaustive catalog of the 340 non-abelian 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).
*
*
*
Further reading
*
*
*
External links
*{{MathWorld, title=p-Group, id=p-Group, author=Rowland, Todd, author-link=Todd Rowland, author2=Weisstein, Eric W., author2-link=Eric W. Weisstein, name-list-style=amp