p-group
In group theory, a p-group (where p is a prime number) is a group in which the order of every element is a power of p.[1] This definition encompasses both finite and infinite groups, with infinite p-groups being those where all elements still satisfy this order condition despite the group's infinite cardinality.[2] For finite groups, the definition is equivalent to the group having order p<sup>n</sup> for some nonnegative integer n.[3] Finite p-groups are fundamental building blocks in the classification of finite groups, particularly through Sylow's theorems, which assert the existence and conjugacy of maximal p-subgroups (Sylow p-subgroups) in any finite group whose order is divisible by p.[3] Every nontrivial finite p-group possesses a nontrivial center Z(G), and in fact, it has normal subgroups of order p<sup>m</sup> for every m with 0 ≤ m ≤ n.[3] All finite p-groups are nilpotent, meaning their lower central series terminates in finitely many steps, and this nilpotency follows from the nontriviality of the center via induction.[3] The structure of finite p-groups becomes increasingly complex as n grows; for example, groups of order p<sup>2</sup> are all abelian and isomorphic to either the cyclic group C<sub>p<sup>2</sup></sub> or the direct product C<sub>p</sub> × C<sub>p</sub>, while higher orders admit non-abelian examples like the dihedral and quaternion groups for p=2.[3] Infinite p-groups, such as the Prüfer p-group (also known as the p<sup>∞</sup>-quasicyclic group), provide examples where every proper subgroup is finite and cyclic of p-power order, illustrating torsion structures without finite overall order.[4] The study of p-groups originated in the 19th century with contributions from mathematicians like Cauchy and Sylow, whose work on solvable groups and prime-power subgroups laid the groundwork for modern finite group theory.[3]Definition and Fundamentals
Definition
In group theory, a p-group, where p is a prime number, is a group G (finite or infinite) in which the order of every element is a power of p.[5] For finite p-groups, this condition is equivalent to the order of G being p<sup>n</sup> for some nonnegative integer n.[5] A basic consequence is that the trivial group, whose sole element has order 1 = p<sup>0</sup>, qualifies as a p-group for every prime p and is the unique group of order 1.[5] More generally, groups where all element orders are powers of primes from a set \pi are termed \pi-groups, with the single-prime case corresponding to \pi = \{p\}.[5] The study of p-groups was advanced by Philip Hall in his 1928 work on solvable groups.[6]Finite versus Infinite p-Groups
A finite p-group is a group whose order is exactly a power of a prime p, denoted as |G| = p^n for some nonnegative integer n.[7] In such groups, the Sylow p-subgroup coincides with the group itself and is therefore normal.[8] A defining property of finite p-groups is that they are nilpotent, meaning their lower central series terminates at the trivial subgroup after finitely many steps; this follows from the nontrivial center of any nontrivial p-group and induction on the order. In contrast, an infinite p-group is an infinite group in which the order of every element is a power of p. While some infinite p-groups are locally finite (every finitely generated subgroup is finite), others are not; for example, there exist finitely generated infinite p-groups. A canonical example is the Prüfer p-group, also known as the quasicyclic p-group or ℤ(p^∞), which is the p-primary component of the quotient group ℚ/ℤ and consists of all p-power roots of unity in the complex numbers under multiplication; it is countable and torsion, with every proper subgroup finite and cyclic.[9] All elements in any p-group, finite or infinite, have order a power of p, ensuring the group is periodic and torsion.[7] A key distinction arises in structural properties: while all finite p-groups are nilpotent, infinite p-groups need not be, though they share the periodic nature of p-groups. Infinite p-groups connect to the Burnside problem, particularly its restricted variant, where solutions show that finitely generated p-groups of bounded exponent p^k are finite, implying any infinite finitely generated p-group must have unbounded exponents.[10] This highlights how infinitude in p-groups often involves unbounded torsion orders, contrasting the controlled finite structure.[11]Core Structural Properties
Non-Abelian Characteristics
A defining feature of non-abelian finite p-groups is that their center Z(G) is a proper, non-trivial subgroup, with |Z(G)| ≥ p.[12] This follows from the class equation for a finite group G of order p^n:|G| = |Z(G)| + \sum |G : C_G(g_i)|,
where the sum runs over representatives g_i of conjugacy classes outside the center, and each centralizer index |G : C_G(g_i)| is a power of p strictly greater than 1.[12] Since |G| is a power of p, the sum must also be a power of p, implying that |Z(G)| is divisible by p.[12] For non-abelian G, Z(G) ≠ G, so the center provides a non-trivial proper normal subgroup that captures the extent to which G fails to be abelian. The derived subgroup G' of a finite p-group G is contained in the Frattini subgroup Φ(G), the intersection of all maximal subgroups of G.[7] Moreover, Φ(G) is generated by G' and the subgroup G^p of p-th powers, so the quotient G/Φ(G) is an elementary abelian p-group—hence abelian of exponent p—and can be viewed as a vector space over the field \mathbb{F}_p.[7] The dimension of this vector space equals the minimal number of generators d(G) of G:
\dim_{\mathbb{F}_p} (G / \Phi(G)) = d(G). [7] This structure highlights how non-abelian p-groups are built from their "linear" quotients modulo the Frattini, with commutators and p-th powers forming the "non-linear" kernel. All finite p-groups, including non-abelian ones, are nilpotent.[13] The nilpotency class, defined as the length of the lower central series minus one, is at most \log_p |G|.[14] This bound arises because the upper central series ascends through non-trivial extensions of order at least p at each step, reaching G in at most \log_p |G| steps.[14] In non-abelian cases, the class is at least 2 but remains controlled by the order, ensuring a finite distance from abelianness.
Automorphism Groups
The automorphism group \operatorname{Aut}(G) of a p-group G consists of all isomorphisms from G to itself, forming a group under composition. A key subgroup is the inner automorphism group \operatorname{Inn}(G), which comprises conjugations by elements of G and is isomorphic to G/Z(G), where Z(G) is the center of G; since Z(G) is nontrivial for non-trivial finite p-groups, \operatorname{Inn}(G) is itself a p-group. The outer automorphism group is the quotient \operatorname{Out}(G) = \operatorname{Aut}(G)/\operatorname{Inn}(G), which encodes symmetries of G modulo inner ones. For finite p-groups, \operatorname{Aut}(G) exhibits rich p-power structure. Any such G admits a faithful action of \operatorname{Aut}(G) on the Frattini quotient G/\Phi(G) \cong (\mathbb{Z}/p\mathbb{Z})^{d(G)}, where \Phi(G) is the Frattini subgroup and d(G) is the minimal number of generators of G; this implies that |\operatorname{Aut}(G)| is divisible by the p-part of |\operatorname{GL}(d(G), p)|, namely p^{d(G)(d(G)-1)/2}. More strikingly, Helleloid and Martin proved that \operatorname{Aut}(G) is itself a p-group for almost all finite p-groups of a given order p^n, in the sense that the proportion of such groups where this holds approaches 1 as n \to \infty.[15] A prominent exception occurs for elementary abelian p-groups G \cong (\mathbb{Z}/p\mathbb{Z})^n, where \operatorname{Aut}(G) \cong \operatorname{GL}(n, p), whose order includes factors coprime to p. In fact, \operatorname{Aut}(G) \cong \operatorname{GL}(n, p) if and only if G is elementary abelian of order p^n.[16] Gaschütz established foundational results on outer automorphisms of finite p-groups, proving that every such G has nontrivial elements in \operatorname{Out}(G), and moreover, if G is not cyclic of order p, then \operatorname{Out}(G) contains an element of p-power order. This implies the existence of outer p-automorphisms, highlighting the p-local nature of symmetries in these groups. Extensions by Schmid show that for nonabelian finite p-groups, such outer automorphisms can act trivially on the center. Regarding automorphism towers—the iterative construction G_0 = G, G_{i+1} = \operatorname{Aut}(G_i)—Gaschütz's insights contribute to understanding stabilization in nilpotent settings, though full resolution for soluble groups relies on later work by Zelmanov resolving the general tower problem affirmatively. For infinite p-groups, the structure of \operatorname{Aut}(G) varies widely; while \operatorname{Out}(G) is often infinite (e.g., for the Prüfer p-group \mathbb{Z}(p^\infty), \operatorname{Aut}(G) \cong \mathbb{Z}_p^\times up to isomorphism, yielding infinite \operatorname{Out}(G) since G is abelian), it can be finite in specific constructions. Notably, every group arises as \operatorname{Out}(H) for some locally finite p-group H, demonstrating the flexibility of outer symmetries even in infinite cases.[17]Key Examples and Constructions
Abelian p-Groups
Abelian p-groups form a fundamental subclass of p-groups, consisting of those where the group operation is commutative. These groups play a crucial role in the structure theory of abelian groups and serve as building blocks for more general classifications, including their appearance as Sylow p-subgroups in finite groups.[18]Finite Case
Finite abelian p-groups admit a complete classification via the fundamental theorem of finite abelian groups, which specializes to the p-primary component. Specifically, every finite abelian p-group G is isomorphic to a direct sum of cyclic groups of p-power order:G \cong \bigoplus_{i=1}^r \mathbb{Z}/p^{k_i}\mathbb{Z},
where k_1 \geq k_2 \geq \cdots \geq k_r > 0. This decomposition into elementary divisors is unique up to isomorphism, with the multiset \{k_1, \dots, k_r\} serving as a complete set of invariants.[18] An alternative presentation uses invariant factors, where G decomposes as a direct sum \mathbb{Z}/p^{m_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/p^{m_s}\mathbb{Z} with m_1 \mid m_2 \mid \cdots \mid m_s. The order of G satisfies |G| = p^{\sum_{i=1}^r k_i}, reflecting the total exponent sum in the elementary divisor form.[18] A key property is the exponent of G, defined as the least common multiple of the orders of its elements, which equals p^{\max\{k_i\}} in the elementary divisor decomposition.[18]
Infinite Case
Infinite abelian p-groups exhibit greater diversity and lack a simple finite-type classification, but they can often be expressed as direct sums or direct products of cyclic p-groups. A canonical example is the Prüfer p-group \mathbb{Z}(p^\infty), which is the direct limit of the system \mathbb{Z}/p^n\mathbb{Z} for n \geq 1 and serves as the injective hull of \mathbb{Z}/p\mathbb{Z} in the category of abelian groups.[19] For countable torsion abelian p-groups, Ulm's theorem provides a classification using Ulm invariants f_\alpha(G) for ordinals \alpha, where f_\alpha(G) counts the dimension of the \alpha-th Ulm factor, determining the isomorphism type uniquely.[20] In general, all abelian p-groups—finite or infinite—are precisely the torsion modules over the ring \mathbb{Z}_p of p-adic integers.[19] The exponent remains well-defined as the lcm of element orders, though it may be infinite.[19]Non-Abelian Examples
Non-abelian p-groups provide essential examples that highlight the departure from commutativity in p-group structures, often arising as semidirect products or matrix groups with non-trivial centers. These groups illustrate key properties such as extraspecial structures and varying exponents, which are central to understanding the diversity of p-groups beyond the abelian case.[21] A fundamental example for p=2 is the dihedral group of order $2^n, denoted D_{2^n}, which consists of symmetries of a regular $2^{n-1}-gon. It has the presentation \langle r, s \mid r^{2^{n-1}} = s^2 = 1, srs^{-1} = r^{-1} \rangle, where r generates rotations and s a reflection, yielding a non-abelian group of order $2^n with a cyclic subgroup of index 2.[22] This construction extends the classical dihedral group and demonstrates how inversion actions produce non-commutativity in 2-groups.[23] Another prominent 2-group is the quaternion group Q_8 of order 8, with presentation \langle x, y \mid x^4 = 1, x^2 = y^2, yxy^{-1} = x^{-1} \rangle. Here, the center is \{1, x^2\}, and all non-central elements have order 4, distinguishing it from other non-abelian groups of order 8.[24] This group exemplifies an extraspecial 2-group, where the center and derived subgroup coincide and have order 2.[7] For odd primes p, the Heisenberg group modulo p, also known as the extraspecial group of exponent p and order p^3, can be realized as the group of upper triangular 3×3 matrices over the finite field \mathbb{F}_p with 1s on the diagonal. Elements are of the form \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, with multiplication yielding order p^3, a center of order p, and quotient by the center isomorphic to (\mathbb{Z}/p\mathbb{Z})^2.[25] This nilpotent group has non-trivial center and captures Heisenberg-like commutation relations in finite settings.[26] In general, all non-abelian groups of order p^3 fall into two isomorphism classes: the Heisenberg group of exponent p, and for odd p, a semidirect product \mathbb{Z}/p^2\mathbb{Z} \rtimes \mathbb{Z}/p\mathbb{Z} of exponent p^2, where the action is multiplication by $1+p. For p=2, the non-abelian groups of order 8 are the dihedral group D_8 and Q_8. These classifications underscore the limited but distinct non-abelian structures at this order.[21] A broader construction for p=2 is the generalized dihedral group over an abelian 2-group A, formed as the semidirect product A \rtimes \mathbb{Z}/2\mathbb{Z} where \mathbb{Z}/2\mathbb{Z} acts by inversion on A. This yields a non-abelian 2-group with A as a normal subgroup of index 2, generalizing the classical dihedral case when A is cyclic. In these examples, non-trivial centers often arise from the kernel of the action, contributing to their nilpotency.[27]Advanced Constructions
One prominent construction of finite p-groups involves wreath products, particularly iterated ones. The regular wreath product \mathbb{Z}_p \wr \mathbb{Z}_p consists of a base group isomorphic to (\mathbb{Z}_p)^p acted upon by a cyclic group of order p, yielding a group of order p^{p+1}.[28] Iterating this process—forming higher wreath powers such as \mathbb{Z}_p \wr (\mathbb{Z}_p \wr \mathbb{Z}_p) and continuing—produces p-groups of exponentially growing order and increasing nilpotency class. These iterated wreath products demonstrate that p-groups exist with arbitrarily large nilpotency class, as starting from a p-group of class at most p and iterating yields groups of unbounded class.[29] Another key construction arises from linear algebra over finite fields. The group UT(n, p) of n × n unitriangular matrices over the field \mathbb{F}_p (with 1s on the diagonal and entries above the diagonal in \mathbb{F}_p) forms a p-group of order p^{n(n-1)/2}, as the superdiagonal and above provide that many independent entries. This group is nilpotent of class exactly n-1, with the lower central series corresponding to the levels of superdiagonals.[30] Extraspecial p-groups provide a canonical family of non-abelian p-groups with controlled structure. An extraspecial p-group G is a non-abelian p-group such that its center Z(G), derived subgroup G', and Frattini subgroup Φ(G) all coincide and have order p, while G/Z(G) is elementary abelian of even rank 2m, giving |G| = p^{2m+1}. For odd p, there are two non-isomorphic extraspecial p-groups of each order p^{2m+1} (m ≥ 1): one of exponent p (the Heisenberg type) and one of exponent p^2 (the semidirect product type). Each family can be realized as central products of the corresponding basic extraspecial group of order p^3, where the Heisenberg group of order p^3 serves as the basic building block for the exponent-p family, and larger groups in each family are obtained by amalgamating centers in a controlled manner. For p=2, the classification involves central products incorporating dihedral and quaternion factors of order 8.[26][31] For infinite p-groups, constructions addressing the Burnside problem yield significant examples. The Golod-Shafarevich theorem provides a criterion for the infinitude of pro-p groups via inequalities on relations in presentations, enabling the explicit construction of infinite discrete p-groups generated by d ≥ 2 elements where every element has order a power of p (p-torsion). For every prime p and d ≥ 2, such an infinite d-generated p-torsion group exists, often realized as quotients of free groups satisfying the Golod-Shafarevich inequality. These groups highlight the existence of infinite p-groups with bounded exponent, contrasting with finite cases.[32]Classification Results
Small Order Classifications
The classification of p-groups begins with the smallest orders, providing foundational examples that illustrate both abelian and non-abelian structures. For order p, where p is prime, there is only one group up to isomorphism: the cyclic group \mathbb{Z}/p\mathbb{Z}, generated by any non-identity element, with presentation \langle x \mid x^p = 1 \rangle.[33] This group is elementary abelian, has exponent p, and nilpotency class 1. For order p^2, there are exactly two groups up to isomorphism, both abelian by the fundamental theorem of finite abelian groups. These are the cyclic group \mathbb{Z}/p^2\mathbb{Z}, with presentation \langle x \mid x^{p^2} = 1 \rangle and exponent p^2, and the elementary abelian group \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}, with presentation \langle x, y \mid x^p = y^p = 1, \, xy = yx \rangle and exponent p. Both have nilpotency class 1.[34] For order p^3, there are five groups up to isomorphism, comprising three abelian and two non-abelian cases; this count holds for odd primes p, while the non-abelian groups differ slightly for p = 2. The abelian groups follow from the fundamental theorem: \mathbb{Z}/p^3\mathbb{Z} (exponent p^3), \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} (exponent p^2), and (\mathbb{Z}/p\mathbb{Z})^3 (exponent p), all with nilpotency class 1.[34] The non-abelian groups are extraspecial p-groups of order p^3, each with center and derived subgroup of order p, quotient by the center isomorphic to (\mathbb{Z}/p\mathbb{Z})^2, and nilpotency class 2. For odd p, one is the Heisenberg group modulo p (also called the extraspecial group of exponent p), with presentation \langle x, y \mid x^p = y^p = [x,y]^p = 1, \, [[x,y],x] = [[x,y],y] = 1 \rangle and exponent p; the other is the semidirect product \mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/p^2\mathbb{Z}, with presentation \langle x, y \mid x^p = 1, \, y^{p^2} = 1, \, yxy^{-1} = x^{1+p} \rangle and exponent p^2. For p = 2 (order 8), the non-abelian groups are the dihedral group D_4 of order 8, with presentation \langle x, y \mid x^4 = y^2 = 1, \, yxy^{-1} = x^{-1} \rangle and exponent 4, and the quaternion group Q_8, with presentation \langle x, y \mid x^4 = 1, \, x^2 = y^2, \, yxy^{-1} = x^{-1} \rangle and exponent 4.[21] The following table summarizes the groups of order p^3, including presentations and key properties:| Group | Presentation | Exponent | Nilpotency Class | Notes |
|---|---|---|---|---|
| \mathbb{Z}/p^3\mathbb{Z} | \langle x \mid x^{p^3} = 1 \rangle | p^3 | 1 | Abelian, cyclic |
| \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} | \langle x, y \mid x^{p^2} = y^p = 1, \, xy = yx \rangle | p^2 | 1 | Abelian |
| (\mathbb{Z}/p\mathbb{Z})^3 | \langle x, y, z \mid x^p = y^p = z^p = 1, \, [x,y] = [x,z] = [y,z] = 1 \rangle | p | 1 | Abelian, elementary |
| Heisenberg mod p (odd p) | \langle x, y \mid x^p = y^p = [x,y]^p = 1, \, [[x,y],x] = [[x,y],y] = 1 \rangle | p | 2 | Non-abelian, extraspecial |
| \mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/p^2\mathbb{Z} (odd p) | \langle x, y \mid x^p = y^{p^2} = 1, \, yxy^{-1} = x^{1+p} \rangle | p^2 | 2 | Non-abelian |
| D_4 (p=2) | \langle x, y \mid x^4 = y^2 = 1, \, yxy^{-1} = x^{-1} \rangle | 4 | 2 | Non-abelian, dihedral |
| Q_8 (p=2) | \langle x, y \mid x^4 = 1, \, x^2 = y^2, \, yxy^{-1} = x^{-1} \rangle | 4 | 2 | Non-abelian, quaternion |