A coprime action is an action of a finite group \(A\) by automorphisms on a finite group \(G\) with \((|A|,|G|)=1\). Coprimeness allows orbit counting, Sylow theory, and averaging to recover fixed-point lifting, invariant complements, and commutator decompositions that fail for general actions. It runs through Schur–Zassenhaus theory, Frobenius groups, Thompson subgroups, and local analysis. [@isaacs2008]

Definition and notation

The fixed-point subgroup and action-commutator subgroup are

\[ C_G(A)=\{g\in G\mid g^a=g\text{ for every }a\in A\}, \]
\[ [G,A]=\langle[g,a]=g^{-1}g^a\mid g\in G,a\in A\rangle. \]

The action may be encoded in the semidirect product \(G\rtimes A\), where the normal kernel and complement have coprime orders.

Invariant Sylow subgroups

For solvable \(G\), the same method applies to any set of primes \(\pi\): there are \(A\)-invariant Hall \(\pi\)-subgroups and they are conjugate by \(C_G(A)\).

Lifting fixed points from quotients

For arbitrary actions only the right-to-left inclusion is automatic; coprimeness is what forces the reverse inclusion.

Commutator decomposition

The identity \([G,A,A]=[G,A]\) is the stability of action commutators: taking a second commutator with the acting group does not shrink the first commutator subgroup.

Centralization criteria

The last implication is immediate from commutator stability and is the common "centralizes twice, hence centralizes" step in local arguments.

Glauberman's lemma

Glauberman's lemma simultaneously explains the existence and fixed-point-group conjugacy of invariant Sylow, Hall, and complement subgroups.

Examples

What fails without coprimeness

Likewise, in module characteristic dividing \(|A|\), the averaging idempotent \(|A|^{-1}\sum a\) does not exist and \(A\)-modules need not be completely reducible.

Main applications

  • existence and conjugacy of complements in Schur–Zassenhaus theory;
  • fixed-point-free action of Frobenius complements and nilpotence of Frobenius kernels;
  • choosing invariant Sylow, Hall, and characteristic subgroups in local analysis;
  • Thompson's \(P\times Q\) lemma, the normal \(J\) theorem, and \(ZJ\) theory;
  • reducing second-level action commutators to first-level commutators;
  • Maschke averaging in finite-group representation theory.

This article follows the coprime-action toolkit developed across Chapters 3, 6, and 7 of Isaacs and supplies proofs of fixed-point lifting, commutator stability, and Glauberman's lemma. Terminology was cross-checked against standard finite-group texts. [@isaacs2008] [@gorenstein1968] [@glauberman1968]