# P-group

A **p-group** is a group in which every non-identity element has order a power of a fixed prime p, that is, each element x satisfies x^(p^n) = 1 for some n depending on the element; p-groups are also called primary groups, and for p = 2, 3, 5 one speaks of 2-groups, 3-groups, and so on.<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup> For finite groups this elementwise definition is equivalent to the order of the group being a power of p, proved via Lagrange's and Cauchy's theorems.<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup> p-groups are the building blocks of finite group theory: the classification of finite nilpotent groups reduces entirely to them.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup>

| Key fact | Detail |
|---|---|
| Definition | Every non-identity element satisfies x^(p^n) = 1 for a fixed prime p; for finite groups, equivalent to the group order being a power of p<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup><sup> • </sup><sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup> |
| Center | The center of a nontrivial finite p-group is nontrivial, proved by the class equation<sup>[4](https://brilliant.org/wiki/p-groups/)</sup> |
| Subgroup structure | Every maximal subgroup is normal of index p, and every proper subgroup is strictly contained in its normalizer<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup> |
| Nilpotence | A group of order p^e is nilpotent; every finite nilpotent group is a direct product of groups of prime-power order<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> |
| Growth of the count | Groups of order 2^n for n = 1 to 10: 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487367289<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> |
| Asymptotics | For n = p^e, f(n) = n^(A mu^2) with A = 2/27 + O(mu^(−1/3)); Pyber extended the matching upper bound to arbitrary n<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> |
| Recent correction | Burrell corrected the Eick–O'Brien enumeration of the groups of order 1024; GAP's Small Groups library version 1.5 lists 423,164,062 of them<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup><sup> • </sup><sup>[5](https://docs.gap-system.org/pkg/smallgrp/doc/chap1.html)</sup> |

## Definition and first examples

The elementwise definition and the order definition agree for finite groups. If |G| = p^k, Lagrange's theorem forces the order of each element to divide p^k, so it is a power of p. Conversely, if every element has p-power order but some prime q ≠ p divided |G|, Cauchy's theorem would produce an element of order q, a contradiction; hence |G| is a power of p.<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup>

Three related terms need separating. A **p-subgroup** is any subgroup that is itself a p-group. A **Sylow p-subgroup** of a finite group G is a p-subgroup of order p^k, where p^k is the largest power of p dividing |G|.<sup>[6](https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/4d34fe81d3457ac23f455f7d53360075_MIT18_703S13_pra_l_13.pdf)</sup> A **pi-group** generalizes the notion by allowing every element to satisfy x^m = 1 where all prime divisors of m lie in a fixed set pi of primes.<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup>

There are five groups of order 8.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup>

## Elementary structural properties

**The nontrivial center.** The center Z(G) of a nontrivial finite p-group is itself nontrivial. The proof runs through the class equation: G decomposes into conjugacy classes, and the size of the class of an element g outside the center equals the index [G : C_G(g)] of its centralizer, which is a power of p greater than 1 and hence divisible by p. Since p also divides |G|, it must divide the remaining term |Z(G)|, so the center cannot be trivial.<sup>[4](https://brilliant.org/wiki/p-groups/)</sup> Iterating the construction of centers produces the upper central series, which reaches G; the lower central series, defined by γ_(i+1)(G) = [G, γ_i(G)], reaches the identity. Both have length c, the nilpotency class of G.<sup>[7](https://users.monash.edu/~heikod/icts2016/CPGlecture1.pdf)</sup> For a group of order p^n the class satisfies 0 ≤ c ≤ n − 1, with c = 0 only for the trivial group and c = 1 exactly when the group is abelian.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup>

**Subgroups of index p.** Every maximal subgroup of a finite p-group is normal and has index p, and every proper subgroup H is strictly contained in its normalizer N_G(H).<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup>

**Simplicity.** A finite simple p-group must be cyclic of order p: its center is nontrivial, so a simple group equals its center and is abelian, and an abelian simple group is cyclic of prime order.<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup> Every p-group is solvable.<sup>[4](https://brilliant.org/wiki/p-groups/)</sup> The nontrivial-center theorem fails for infinite p-groups.<sup>[2](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)</sup>

## Role in finite group theory

Sylow's theorems place p-groups at the center of finite group theory. For a finite group G and a prime p dividing |G|, Sylow p-subgroups exist; any two of them are conjugate; every p-subgroup is contained in one of them; and their number n_p satisfies n_p ≡ 1 (mod p) and n_p divides |G|/p^k.<sup>[9](https://math.dartmouth.edu/~trs/PreTeXtProjects/css-playground/section-9.html)</sup><sup> • </sup><sup>[10](https://sporadic.stanford.edu/bump/group/gr1_6.html)</sup> The congruence and divisibility conditions give a practical criterion for deciding when a [Sylow p-subgroup](https://www.edgepedia.org/sylow-p-subgroup) is normal, a step used throughout the program of classifying finite simple groups.<sup>[9](https://math.dartmouth.edu/~trs/PreTeXtProjects/css-playground/section-9.html)</sup> The normalizer of a nontrivial p-subgroup is called a local subgroup, and local subgroups are central to the study of finite simple groups and to modular representation theory.<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup>

**Nilpotence.** A group of prime-power order is nilpotent, and conversely every finite nilpotent group is a direct product of groups of prime-power order, so the classification of finite nilpotent groups reduces completely to p-groups.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> Equivalently, a nilpotent group is the direct product of its Sylow p-subgroups, and every finite group has a unique maximal normal nilpotent subgroup, the Fitting subgroup F(G).<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup>

## Classification and enumeration

The workhorse is the **p-group generation algorithm** of Eamonn O'Brien. The algorithm determines the immediate descendants of a given p-group G up to isomorphism, using the lower p-central series λ_1(P) = P, λ_(i+1)(P) = [λ_i(P), P] λ_i(P)^p, and constructing the p-cover, an extension related to the p-multiplicator; given a description of the automorphism group of G, it outputs all immediate descendants together with their automorphism groups.<sup>[11](https://www.sciencedirect.com/science/article/pii/S074771710880082X?via%3Dihub)</sup><sup> • </sup><sup>[12](https://gap-packages.github.io/sglppow/htm/CHAP003.htm)</sup><sup> • </sup><sup>[13](https://users.monash.edu/~heikod/icts2016/CPGlecture3.pdf)</sup> Each non-elementary-abelian p-group is an immediate descendant of the quotient by the last nontrivial term of that series, so the algorithm organizes all p-groups of a given order into a tree.<sup>[12](https://gap-packages.github.io/sglppow/htm/CHAP003.htm)</sup>

The algorithm's results feed the **Small Groups library** in GAP and Magma, built on work of Besche, Eick, and O'Brien. The library contains all groups of order p^n for n ≤ 6 and all primes p, the 907,489 groups of order p^7 for p = 3, 5, 7, 11, a subset of 423,164,062 of the 49,487,367,289 groups of order 1024, and 395,703 cubefree groups of order at most 50,000.<sup>[5](https://docs.gap-system.org/pkg/smallgrp/doc/chap1.html)</sup><sup> • </sup><sup>[14](https://magma.maths.usyd.edu.au/magma/handbook/text/815)</sup><sup> • </sup><sup>[27](https://msp.org/ant/2023/17-2/ant-v17-n2-p03-p.pdf)</sup> The groups of order 512 were first enumerated and later determined explicitly by p-group generation.<sup>[15](https://github.com/gap-packages/smallgrp/blob/master/README)</sup> O'Brien's algorithm determined the groups of orders 128 and 256, the latter with 56,092 groups, a computation beyond hand calculation in 1990, and it determined the 1,396,077 groups of order 3^8.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> A complementary cohomological algorithm using the Cauchy–Frobenius lemma independently established the 10,494,213 groups of order 2^9.<sup>[16](https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/enumerating-pgroups/8FEA80118A2945650942077DB7BEFE2C)</sup>

**Asymptotics.** For n = p^e with mu = e, Higman and Sims showed that the number f(n) of groups of order n satisfies f(n) = n^(A mu^2) with A = 2/27 + O(mu^(−1/3)); Pyber extended the upper bound f(n) ≤ n^(B mu^2) with B ≤ 2/27 + o(1) to arbitrary n.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> In exponential form, log_p f(p^m) = 2m^3/27 + O(m^(5/2)).<sup>[17](https://arxiv.org/html/2610.11092)</sup> Higman also conjectured that for fixed n the number of groups of order p^n is a PORC function of p, meaning polynomial on residue classes, and the count is known to be PORC for n ≤ 7; O'Brien and Vaughan-Lee determined the groups of order dividing p^7 for primes p > 5 and showed the count is PORC.<sup>[18](https://www.sciencedirect.com/science/article/pii/S0021869318301716)</sup><sup> • </sup><sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> For a family of groups, the number of isomorphism classes of order p^n with central Frattini subgroup is PORC in p, extending Higman's 1960 theorem.<sup>[19](https://arxiv.org/abs/0710.0394)</sup>

## By the numbers

The counts grow explosively with the exponent. For orders 2^n with n = 1 to 10 the numbers of isomorphism types are 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, and 49487367289.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> The count depends on the prime structure of n more than on its size: there are 14 groups of order 16 but only one group of order 17, so f(n) is not monotonic.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> The same pattern appears at 4, 8, 16, and 32, with g(4) = 2, g(8) = 5, g(16) = 14, and g(32) = 51.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup>

**Almost all groups are p-groups.** As n tends to infinity, the probability that a group of order at most n chosen uniformly at random is a p-group tends to 1; this was long a folklore conjecture in the field.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup><sup> • </sup><sup>[7](https://users.monash.edu/~heikod/icts2016/CPGlecture1.pdf)</sup>

## Important classes of p-groups

**Extraspecial groups.** An extraspecial p-group is one for which G/Z(G) is elementary abelian and Z(G) has order p.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup> In such a group the center, the commutator subgroup, and the Frattini subgroup all coincide and equal a single cyclic group of order p, the group has nilpotency class 2, and for each prime p and each n ≥ 1 there are exactly two extraspecial groups of order p^(1+2n) up to isomorphism.<sup>[20](https://finitegroupdatabase.com/c/extraspecial)</sup> Extraspecial groups appear frequently in representation theory.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup>

**Groups of exponent p.** For p > 7, Vaughan-Lee determined the groups of order p^8 and exponent p, finding the count p^4 + 2p^3 + 20p^2 + 147p + (3p + 29)gcd(p−1, 3) + 5gcd(p−1, 4) + 1246.<sup>[21](https://users.ox.ac.uk/~vlee/PORC/p8expp.pdf)</sup>

**Coclass theory.** The coclass of a p-group of order p^n and nilpotency class c is n − c; groups of coclass 1 are exactly the groups of maximal class.<sup>[22](https://link.springer.com/article/10.1007/s00013-023-01867-w)</sup><sup> • </sup><sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup> In 1980 Charles Leedham-Green and Michael F. Newman introduced the invariant and proposed five far-reaching conjectures; all turned out to be true, and their proofs created a new structure theory of p-groups.<sup>[22](https://link.springer.com/article/10.1007/s00013-023-01867-w)</sup> Coclass Conjecture A, for example, states that for each prime p and each r there is a bound f such that every finite p-group of coclass r has a normal subgroup of class 2 with index at most p^f.<sup>[22](https://link.springer.com/article/10.1007/s00013-023-01867-w)</sup> After the conjectures were proved, the pro-2-groups of coclass at most 3 were determined and used to classify the 2-groups of coclass at most 3 into families.<sup>[23](https://www.ams.org/journals/tran/1999-351-01/S0002-9947-99-02124-8/)</sup>

**Open conjectures.** A 2022 survey collects conjectures that remain unresolved in finite p-group theory, including Hua-Tuan's conjecture, Higman's conjecture, Oliver's conjecture, and Wiegold's conjecture.<sup>[24](https://journal.hep.com.cn/fmc/EN/10.1007/s11464-022-1001-6)</sup> The 2024 classification survey lists determining the groups of order p^8 among its open problems.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup>

## Generalizations and infinite p-groups

Beyond pi-groups, the theory extends in several directions. Locally finite p-groups, infinite groups in which every finitely generated subgroup is a finite p-group, are never simple: there exist locally finite p-groups without nontrivial normal abelian subgroups, and even locally finite p-groups that coincide with their commutator subgroup.<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup> On the finitary side, the restricted Burnside problem, which bounds the size of finitely generated residually finite p-groups of bounded exponent, was solved by E. I. Zel'manov in 1989.<sup>[1](https://encyclopediaofmath.org/wiki/P-group)</sup> Computationally, every finite p-group can be associated with a Lie ring.<sup>[8](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)</sup> O'Brien and Vaughan-Lee's classification of the groups of order p^7 proceeds by classifying nilpotent Lie rings of order p^7 via the [Baker–Campbell–Hausdorff formula](https://www.edgepedia.org/baker-campbell-hausdorff-formula).<sup>[12](https://gap-packages.github.io/sglppow/htm/CHAP003.htm)</sup>

## Recent developments

**The order-1024 correction.** Burrell found a mistake in the Eick–O'Brien enumeration of the groups of order 1024, confirmed the rest of that enumeration, and made some groups of order 1024 available as a GAP library; Small Groups library version 1.5 corrected the count accordingly.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup><sup> • </sup><sup>[5](https://docs.gap-system.org/pkg/smallgrp/doc/chap1.html)</sup> The Small Groups library, however, still lists the older count of 423,164,062 groups of order 1024, whereas there are in fact 49,487,367,289 such groups up to isomorphism.<sup>[5](https://docs.gap-system.org/pkg/smallgrp/doc/chap1.html)</sup><sup> • </sup><sup>[27](https://msp.org/ant/2023/17-2/ant-v17-n2-p03-p.pdf)</sup>

**Almost all groups are 2-groups of class two.** A recent preprint proves that almost all finite groups, counted by isomorphism type up to order x, are 2-groups of nilpotency class two: uniformly for 2^m ≤ x < 2^(m+1), all but an O(2^(−cm^2)) proportion have order 2^m and class two. The same work refines the asymptotics, giving log_p f(p^m) = 2m^3/27 + O(m^(5/2)) and, for each fixed prime p, a formula for the number of groups of order p^m with relative error O(p^(−m/3)).<sup>[17](https://arxiv.org/html/2610.11092)</sup>

**Computation at the frontier.** The total number of groups of order 2048 remains unknown, but the number with lower p-central series of length 2 is f_2(2048) = 1,774,274,116,992,170, a lower bound on the full count.<sup>[3](https://link.springer.com/article/10.1007/s10208-024-09688-1)</sup> New work introduces siblings and twins, pairs of non-isomorphic p-groups that are difficult to distinguish by invariants, exhibiting 56 such pairs among the groups of order 2^9 and devising an effective algorithm that solves the group identification problem for all 10,494,213 groups of that order.<sup>[25](https://arxiv.org/abs/2603.09363v1)</sup> On a neighboring problem, the number gnu(n) of groups of cubefree order is now bounded by gnu(n) ≤ n^(2+o(1)), improving the previously recorded bound gnu(n) < n^8.<sup>[26](https://arxiv.org/html/2608.18815v1)</sup>

## References

1. [P-group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/P-group)
2. [pGroups, lecture notes, University of Minnesota](https://www-users.cse.umn.edu/~webb/oldteaching/Year15-16/pGroups.pdf)
3. [Classification of Finite Groups: Recent Developments and Open Problems, Springer (2024)](https://link.springer.com/article/10.1007/s10208-024-09688-1)
4. [P-groups, Brilliant Math & Science Wiki](https://brilliant.org/wiki/p-groups/)
5. [GAP Small Groups Library, Chapter 1](https://docs.gap-system.org/pkg/smallgrp/doc/chap1.html)
6. [18.703 Modern Algebra: Sylow Theorems and Applications, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/4d34fe81d3457ac23f455f7d53360075_MIT18_703S13_pra_l_13.pdf)
7. [Computational Aspects of Finite p-Groups, lecture 1, Bettina Eick](https://users.monash.edu/~heikod/icts2016/CPGlecture1.pdf)
8. [The Theory of p-Groups, D. A. Craven, lecture notes](https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf)
9. [Some Structure and Classification Theorems, Dartmouth](https://math.dartmouth.edu/~trs/PreTeXtProjects/css-playground/section-9.html)
10. [Sylow Theorems, Daniel Bump, Stanford](https://sporadic.stanford.edu/bump/group/gr1_6.html)
11. [The p-Group Generation Algorithm, Journal of Symbolic Computation](https://www.sciencedirect.com/science/article/pii/S074771710880082X?via%3Dihub)
12. [Small Groups Library: Organization of the Data, GAP package sglppow](https://gap-packages.github.io/sglppow/htm/CHAP003.htm)
13. [p-Group Generation, lecture 3, Bettina Eick](https://users.monash.edu/~heikod/icts2016/CPGlecture3.pdf)
14. [Magma Handbook: Small Groups Library](https://magma.maths.usyd.edu.au/magma/handbook/text/815)
15. [gap-packages/smallgrp README](https://github.com/gap-packages/smallgrp/blob/master/README)
16. [Enumerating p-Groups, Journal of the Australian Mathematical Society](https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/enumerating-pgroups/8FEA80118A2945650942077DB7BEFE2C)
17. [Almost All Finite Groups Are 2-Groups of Class Two, arXiv](https://arxiv.org/html/2610.11092)
18. [The Enumeration of Groups of Order p^n q for n ≤ 5, Journal of Algebra](https://www.sciencedirect.com/science/article/pii/S0021869318301716)
19. [Higman's PORC Conjecture for a Family of Groups, arXiv](https://arxiv.org/abs/0710.0394)
20. [Extraspecial Groups, Finite Group Database](https://finitegroupdatabase.com/c/extraspecial)
21. [Groups of Order p^8 and Exponent p, Vaughan-Lee](https://users.ox.ac.uk/~vlee/PORC/p8expp.pdf)
22. [The Origins of Coclass Theory, Archiv der Mathematik](https://link.springer.com/article/10.1007/s00013-023-01867-w)
23. [2-Groups of Coclass at Most 3, Transactions of the American Mathematical Society](https://www.ams.org/journals/tran/1999-351-01/S0002-9947-99-02124-8/)
24. [Some Unsolvable Conjectures in Finite p-Groups, Frontiers of Mathematics in China](https://journal.hep.com.cn/fmc/EN/10.1007/s11464-022-1001-6)
25. [Siblings and Twins in Finite p-Groups and a Group Identification for the Groups of Order 2^9, arXiv](https://arxiv.org/abs/2603.09363v1)
26. [The Number of Groups of Cubefree Order, arXiv](https://arxiv.org/html/2608.18815v1)
27. [Ant V17 N2 P03 P.pdf, msp.org](https://msp.org/ant/2023/17-2/ant-v17-n2-p03-p.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Finite groups and classification › Subgroup structure and Sylow theory*

*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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