A p-group is a finite group of prime-power order. Such groups are the basic local objects of finite group theory: for each prime, a Sylow \(p\)-subgroup carries the largest \(p\)-part of a general finite group. Nontrivial centers, abundant normal subgroups, and nilpotence make induction particularly effective for \(p\)-groups. [@isaacs2008]

Definition and equivalent form

Lagrange's theorem implies that every element of a finite \(p\)-group has prime-power order. Conversely, if every element of a finite group has \(p\)-power order and another prime \(q\) divided the group order, Cauchy's theorem would produce an element of order \(q\). Thus, for finite groups, prime-power group order is equivalent to every element having \(p\)-power order.

For infinite groups the second condition is commonly used as the definition, but many conclusions below are specifically finite.

Counting modulo p

If a finite \(p\)-group \(P\) acts on a finite set \(X\), every orbit has \(p\)-power size. For the global fixed-point set \(X^P\),

\[ |X|\equiv|X^P|\pmod p. \]

Acting on cosets, conjugacy classes, or families of subgroups repeatedly turns this congruence into fixed points and normal subgroups.

Nontrivial center

Consequences include:

  • every minimal normal subgroup of a finite \(p\)-group is central and has order \(p\);
  • even a nonabelian finite \(p\)-group has at least \(p\) central elements;
  • \(\operatorname{Inn}(P)\cong P/Z(P)\) has strictly smaller order, enabling induction.

The normalizer condition

Thus a proper subgroup can always be enlarged through its normalizer. Iterating normalizers eventually reaches \(P\). This is also an important characterization of finite nilpotent groups.

Subgroups of every prime-power order

In particular, one can construct

\[ 1=P_0\triangleleft P_1\triangleleft\cdots\triangleleft P_n=P, \qquad |P_i:P_{i-1}|=p. \]

Maximal and Frattini subgroups

Every maximal subgroup \(M\) of a finite \(p\)-group has index \(p\) and is normal. The intersection of all maximal subgroups is the Frattini subgroup \(\Phi(P)\). It satisfies

\[ \Phi(P)=P'P^p, \]

and \(P/\Phi(P)\) is elementary abelian. Burnside's basis theorem states that elements generate \(P\) precisely when their images generate this vector space. Hence

\[ d(P)=\dim_{\mathbb F_p}P/\Phi(P) \]

is the minimum number of generators.

Nilpotence

Every finite \(p\)-group is therefore solvable as well, but need not be abelian; its nilpotency class and derived length can be large.

Examples

Small orders

  • The only group of order \(p\) is \(C_p\).
  • Groups of order \(p^2\) are abelian: \(C_{p^2}\) and \(C_p\times C_p\).
  • For odd \(p\), there are three abelian and two nonabelian groups of order \(p^3\); \(UT_3(p)\) is one nonabelian example.
  • There are five groups of order eight: three abelian groups, \(D_8\), and \(Q_8\).

The number of groups of order \(p^n\) grows extremely quickly. Modern theory therefore organizes them by nilpotency class, coclass, exponent, rank, or isoclinism rather than by complete lists at large orders.

Role in finite group theory

  • The Sylow theorems reduce the prime-local structure of a finite group to \(p\)-groups.
  • The largest normal \(p\)-subgroup \(O_p(G)\), normalizers, and centralizers are the basic data of \(p\)-local analysis.
  • Transfer, normal p-complements, and fusion control depend on the internal structure of Sylow subgroups.
  • Elementary abelian and extraspecial groups connect local group theory to finite-field linear algebra and symplectic geometry.
  • The classification of finite simple groups makes extensive use of Sylow \(2\)-subgroups and other local subgroups.

Common errors

The English and Chinese Wikipedia articles were used to cross-check examples, classification facts, and modern applications. The structural theorems and proofs were rewritten from Isaacs, Chapter 1. The cited revisions are attributed under CC BY-SA. [@wikipedia-p-group-zh][@wikipedia-p-group-en]