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{{Short description|Group in which the order of every element is a power of p}}
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{{Group theory sidebar |Finite}}
{{DISPLAYTITLE:{{mvar|p}}-group}}
In [[mathematics]], specifically [[group theory]], given a [[prime number]] ''p'', a '''''p''-group''' is a [[Group (mathematics)|group]] in which the [[Order of a group element|order]] of every element is a [[Power (mathematics)|power]] of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a [[nonnegative integer]] ''n'' such that the product of ''p<sup>n</sup>'' copies of ''g'', and not fewer, is equal to the [[identity element]]. The orders of different elements may be different powers of ''p''.

[[Abelian group|Abelian]] ''p''-groups are also called '''''p''-primary''' or simply '''primary'''.

A [[finite group]] is a ''p''-group if and only if its [[Order of a group|order]] (the number of its elements) is a power of ''p''. Given a finite group ''G'', the [[Sylow theorems]] guarantee the existence of a [[subgroup]] of ''G'' of order ''p<sup>n</sup>'' for every [[prime power]] ''p<sup>n</sup>'' that divides the order of ''G''. 

Every finite ''p''-group is [[Nilpotent group|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 group|simple]] ''p''-group, see [[Tarski monster group]].

==Properties==

Every ''p''-group is [[periodic group|periodic]] since by definition every element has [[finite order]].

If ''p'' is prime and ''G'' is a group of order ''p''<sup>''k''</sup>, then ''G'' has a normal subgroup of order ''p''<sup>''m''</sup> for every 1 ≤ ''m'' ≤ ''k''.  This follows by induction, using [[Cauchy's theorem (group theory)|Cauchy's theorem]] and the [[Correspondence theorem (group theory)|Correspondence Theorem]] for groups. A proof sketch is as follows: because the [[center of a group|center]] ''Z'' of ''G'' is [[Trivial group|non-trivial]] (see below), according to [[Cauchy's theorem (group theory)|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]] is that the [[Center (group theory)|center]] of a non-trivial finite ''p''-group cannot be the trivial subgroup.<ref>[[Conjugacy class#Example|proof]]</ref>

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]] 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 group|nilpotent]].

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 group|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 [[Center (group theory)#Higher centers|second center]] and these groups begin the [[upper central series]]. Generalizing the earlier comments about the socle, a finite ''p''-group with order ''p<sup>n</sup>'' contains normal subgroups of order ''p<sup>i</sup>'' with 0 ≤ ''i'' ≤ ''n'', and any normal subgroup of order ''p<sup>i</sup>'' is contained in the ''i''th center ''Z''<sub>''i''</sub>. If a normal subgroup is not contained in ''Z''<sub>''i''</sub>, then its intersection with ''Z''<sub>''i''+1</sub> has size at least ''p''<sup>''i''+1</sup>.

===Automorphisms===
The [[group automorphism|automorphism]] 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]]<!-- Gaschuetz. In fact people tried to figure out how big it was, conjecture, minimal counterexample of order p^6, yada yada-->. 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 [[William Burnside|Burnside]], who showed that the kernel of this map is a ''p''-group. <!-- give corresponding theorems for action on Ω and the socle -->

==Examples==
''p''-groups of the same order are not necessarily [[isomorphism|isomorphic]]; for example, the [[cyclic group]] ''C''<sub>4</sub> and the [[Klein four-group]] ''V''<sub>4</sub> are both 2-groups of order 4, but they are not isomorphic.

Nor need a ''p''-group be [[abelian group|abelian]]; the [[dihedral group]] Dih<sub>4</sub> of order 8 is a non-abelian 2-group. However, every group of order ''p''<sup>2</sup> is abelian.<ref group="note">To prove that a group of order ''p''<sup>2</sup> 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: <math>G=C_{p^2}</math>), 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 <math>p^2</math>) but they commute since ''g'' is central, so the group is abelian, and in fact <math>G=C_p \times C_p.</math></ref>

The dihedral groups are both very similar to and very dissimilar from the [[quaternion group]]s and the [[semidihedral group]]s. Together the dihedral, semidihedral, and quaternion groups form the 2-groups of [[maximal class]], that is those groups of order 2<sup>''n''+1</sup> 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''&nbsp;+&nbsp;1). Then ''W''(''n'') is the Sylow ''p''-subgroup of the [[symmetric group]] Sym(''p''<sup>''n''</sup>). Maximal ''p''-subgroups of the general linear group GL(''n'','''Q''') are direct products of various ''W''(''n''). It has order ''p''<sup>''k''</sup> where ''k''&nbsp;=&nbsp;(''p''<sup>''n''</sup>&nbsp;−&nbsp;1)/(''p''&nbsp;−&nbsp;1). It has nilpotency class ''p''<sup>''n''−1</sup>, 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''<sup>''n''</sup>. The second such group, ''W''(2), is also a ''p''-group of maximal class, since it has order ''p''<sup>''p''+1</sup> and nilpotency class ''p'', but is not a [[regular p-group|regular ''p''-group]]. Since groups of order ''p''<sup>''p''</sup> are always regular groups, it is also a minimal such example.

===Generalized dihedral groups===
When ''p''&nbsp;=&nbsp;2 and ''n''&nbsp;=&nbsp;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''&nbsp;=&nbsp;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<sup>''n''</sup>, but that requires a bit more setup. Let ζ denote a primitive ''p''th root of unity in the complex numbers, let '''Z'''[ζ] be the ring of [[ring of integers|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''<sup>''n''</sup> are normal subgroups of ''E''(''p''), and the example groups are ''E''(''p'',''n'')&nbsp;=&nbsp;''E''(''p'')/''P''<sup>''n''</sup>. ''E''(''p'',''n'') has order ''p''<sup>''n''+1</sup> and nilpotency class ''n'', so is a ''p''-group of maximal class. When ''p''&nbsp;=&nbsp;2, ''E''(2,''n'') is the dihedral group of order 2<sup>''n''</sup>. When ''p'' is odd, both ''W''(2) and ''E''(''p'',''p'') are irregular groups of maximal class and order ''p''<sup>''p''+1</sup>, 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''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub> } and define ''V''<sub>''i''</sub> to be the vector space generated by { ''e''<sub>''i''</sub>, ''e''<sub>''i''+1</sub>, ..., ''e''<sub>''n''</sub> } for 1 ≤ ''i'' ≤ ''n'', and define ''V''<sub>''i''</sub> = 0 when ''i'' &gt; ''n''. For each 1 ≤ ''m'' ≤ ''n'', the set of invertible linear transformations of ''V'' which take each ''V''<sub>''i''</sub> to ''V''<sub>''i''+''m''</sub> form a subgroup of Aut(''V'') denoted ''U''<sub>''m''</sub>. If ''V'' is a vector space over '''Z'''/''p'''''Z''', then ''U''<sub>1</sub> is a Sylow ''p''-subgroup of Aut(''V'') = GL(''n'', ''p''), and the terms of its [[lower central series]] are just the ''U''<sub>''m''</sub>. In terms of matrices, ''U''<sub>''m''</sub> are those upper triangular matrices with 1s one the diagonal and 0s on the first ''m''−1 superdiagonals. The group ''U''<sub>1</sub> has order ''p''<sup>''n''·(''n''−1)/2</sup>, nilpotency class ''n'', and exponent ''p''<sup>''k''</sup> where ''k'' is the least integer at least as large as the base ''p'' [[logarithm]] of ''n''.

==Classification==
The groups of order ''p''<sup>''n''</sup> for 0 ≤ ''n'' ≤ 4 were classified early in the history of group theory,<ref>{{harv|Burnside|1897}}</ref> and modern work has extended these classifications to groups whose order divides ''p''<sup>7</sup>, 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.<ref>{{harv|Leedham-Green|McKay|2002|p=214}}</ref> For example, [[Marshall Hall (mathematician)|Marshall Hall Jr.]] and James K. Senior classified groups of order 2<sup>''n''</sup> for ''n'' ≤ 6 in 1964.<ref>{{Harv|Hall Jr.|Senior|1964}}</ref>

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.<ref>{{harv|Hall|1940}}</ref>

An entirely different method classifies finite ''p''-groups by their '''[[coclass]]''', that is, the difference between their [[composition series|composition length]] and their [[nilpotent group|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 to [[Lie algebra]]s and [[powerful p-group]]s.<ref>{{harv|Leedham-Green|McKay|2002}}</ref> 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 [[Descendant tree (group theory)#Multifurcation and coclass graphs|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''<sup>5</sup> is [[Metabelian group|metabelian]].<ref name="metabelian">{{cite web | url=https://math.stackexchange.com/q/124010/178864 | title=Every group of order ''p''<sup>5</sup> is metabelian | date=24 March 2012 | publisher=Stack Exchange |access-date=7 January 2016}}</ref>

===Up to ''p''<sup>3</sup>===
The trivial group is the only group of order one, and the cyclic group C<sub>''p''</sub> is the only group of order ''p''. There are exactly two groups of order ''p''<sup>2</sup>, both abelian, namely C<sub>''p''<sup>2</sup></sub> and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. For example, the cyclic group C<sub>4</sub> and the [[Klein four-group]] ''V''<sub>4</sub> which is C<sub>2</sub>&nbsp;×&nbsp;C<sub>2</sub> are both 2-groups of order 4.

There are three abelian groups of order ''p''<sup>3</sup>, namely C<sub>''p''<sup>3</sup></sub>, C<sub>''p''<sup>2</sup></sub>&nbsp;×&nbsp;C<sub>''p''</sub>, and C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub>. There are also two non-abelian groups.

For ''p''&nbsp;≠&nbsp;2, one is a semi-direct product of C<sub>''p''</sub>&nbsp;×&nbsp;C<sub>''p''</sub> with C<sub>''p''</sub>, and the other is a semi-direct product of C<sub>''p''<sup>2</sup></sub> with C<sub>''p''</sub>. 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#Heisenberg group modulo an odd prime p|Heisenberg group mod ''p'']].

For ''p''&nbsp;=&nbsp;2, both the semi-direct products mentioned above are isomorphic to the [[dihedral group]] Dih<sub>4</sub> of order 8. The other non-abelian group of order 8 is the [[quaternion group]] Q<sub>8</sub>.

==Prevalence==

=== Among groups ===
The number of isomorphism classes of groups of order ''p<sup>n</sup>'' grows as <math>p^{\frac{2}{27}n^3+O(n^{8/3})}</math>, and these are dominated by the classes that are two-step nilpotent.<ref>{{harv|Sims|1965}}</ref> Because of this rapid growth, there is a [[Mathematical folklore|folklore]] conjecture asserting that almost all [[finite group]]s are 2-groups: the fraction of [[isomorphism class]]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, {{gaps|49|487|367|289}}, or just over 99%, are 2-groups of order 1024.<ref name="Burrell">{{cite journal |last1=Burrell |first1=David |title=On the number of groups of order 1024 |journal=Communications in Algebra |date=2021-12-08 |volume=50 |issue=6 |pages=2408–2410 |doi=10.1080/00927872.2021.2006680 |url=https://www.tandfonline.com/doi/full/10.1080/00927872.2021.2006680}}</ref>

=== 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 (group theory)|Cauchy's theorem]]. In fact, it contains a ''p''-group of maximal possible order: if <math>|G|=n=p^km</math> where ''p'' does not divide ''m,'' then ''G'' has a subgroup ''P'' of order <math>p^k,</math> 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]]. ''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|''p''-core]] <math>O_p(G)</math> (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|''p''-residual subgroup]] <math>O^p(G).</math> 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 [[normalizer]]s of non-identity ''p''-subgroups.<ref>{{harv|Glauberman|1971}}</ref>

The large [[elementary abelian group|elementary abelian subgroup]]s of a finite group exert control over the group that was used in the proof of the [[Feit–Thompson theorem]]. Certain [[Group extension#Central extension|central extension]]s of elementary abelian groups called [[extraspecial group]]s help describe the structure of groups as acting on [[symplectic vector space]]s.

[[Richard Brauer]] classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and [[John Walter (mathematician)|John Walter]], [[Daniel Gorenstein]], [[Helmut Bender]], [[Michio Suzuki (mathematician)|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]]<!-- include the power commutator structure -->

==Footnotes==

===Notes===
{{Reflist|group=note}}

===Citations===
{{Reflist}}

==References==
{{Refbegin}}
*{{Citation | last1=Besche | first1=Hans Ulrich | last2=Eick | first2=Bettina | last3=O'Brien | first3=E. A. | title=A millennium project: constructing small groups | mr=1935567 | year=2002 | journal=International Journal of Algebra and Computation | volume=12 | issue=5 | pages=623–644 | doi=10.1142/S0218196702001115| s2cid=31716675 }}
*{{Citation | last1=Burnside | first1=William | author1-link=William Burnside | title=Theory of groups of finite order | publisher=[[Cambridge University Press]] | year=1897 | isbn=9781440035456 | url=https://books.google.com/books?id=yBgPAAAAIAAJ}}
*{{Citation | last1=Glauberman | first1=George | author1-link=George Glauberman | title=Finite simple groups (Proc. Instructional Conf., Oxford, 1969) | publisher=[[Academic Press]] | location=Boston, MA | mr=0352241 | year=1971 | chapter=Global and local properties of finite groups | pages=1–64}}
*{{citation | last1=Hall Jr. | first1=Marshall | author-link1 = Marshall Hall (mathematician) | last2=Senior | first2=James K. | title=The Groups of Order 2''<sup>n</sup>'' (''n'' &le; 6) | publisher=Macmillan | location=London | year=1964 | mr=168631 | lccn=64016861}} &mdash; An exhaustive catalog of the 340 non-abelian groups of order dividing 64 with detailed tables of defining relations, constants, and [[Lattice of subgroups|lattice]] presentations of each group in the notation the text defines. "Of enduring value to those interested in [[finite groups]]" (from the preface).
*{{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The classification of prime-power groups | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00217491X | doi=10.1515/crll.1940.182.130 | mr=0003389 | year=1940 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1940 | issue=182 | pages=130–141| s2cid=122817195 }}
*{{Citation | last1=Leedham-Green | first1=C. R. | author1-link=Charles Leedham-Green | last2=McKay | first2=Susan | title=The structure of groups of prime power order | publisher=[[Oxford University Press]] | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853548-5 | mr=1918951 | year=2002 | volume=27}}
*{{Citation | last1=Sims | first1=Charles | title=Enumerating p-groups | mr=0169921 | year=1965 | journal=Proc. London Math. Soc. |series=Series 3 | volume=15 | pages=151–166 | doi=10.1112/plms/s3-15.1.151}}
{{Refend}}

==Further reading==
{{Refbegin}}
*{{Citation | last=Berkovich | first=Yakov | author-link=Yakov Berkovich | year=2008 | title=Groups of Prime Power Order | publisher=Walter de Gruyter GmbH | location=Berlin | series=de Gruyter Expositions in Mathematics 46 | volume=1 | isbn=978-3-1102-0418-6}}
*{{Citation | last1=Berkovich | first1=Yakov | last2=Janko | first2=Zvonimir | author2-link=Zvonimir Janko | year=2008 | title=Groups of Prime Power Order | publisher=Walter de Gruyter GmbH | location=Berlin | series=de Gruyter Expositions in Mathematics 47 | volume=2 | isbn=978-3-1102-0419-3}}
*{{Citation | last1=Berkovich | first1=Yakov | last2=Janko | first2=Zvonimir | date=2011-06-16 | title=Groups of Prime Power Order | publisher=Walter de Gruyter GmbH | location=Berlin | series=de Gruyter Expositions in Mathematics 56 | volume=3 | isbn=978-3-1102-0717-0}}
{{Refend}}

==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}}

[[Category:P-groups| ]]