Glauberman's lemma upgrades ordinary transitivity to transitivity by a fixed-point subgroup. It is the common tool for selecting invariant Sylow, Hall, and complement subgroups under coprime action. [@isaacs2008]
Definition
The condition \((|A|,|G|)=1\) concerns the two acting group orders; it does not require \(|A|\) to be coprime to \(|\Omega|\).
Core result
Write \(\Omega\) as a coset space \(G/H\) and interpret an \(A\)-fixed coset as an inclusion of complements in \(G\rtimes A\). Schur–Zassenhaus conjugacy adjusts a coset representative into \(C_G(A)\). The coprime-action core article contains the full proof.
Structural properties
- Applied to the conjugacy orbit of Sylow subgroups, it gives an \(A\)-invariant Sylow subgroup.
- Any two \(A\)-invariant Sylow subgroups are conjugate by \(C_G(A)\).
- For solvable groups it similarly supplies invariant Hall subgroups and fixed-point conjugacy.
- Applied to a set of complements, it produces invariant complements and controlled conjugacy.
- The lifting formula \(C_{G/N}(A)=C_G(A)N/N\) is the internal group version of the same coprime mechanism.
The lemma says that if the whole group can move between two fixed objects, elements centralizing the acting group already suffice.
Example and boundary
Knowledge network
The coprime-action core article proves the lemma and fixed-point lifting, while Sylow and Schur–Zassenhaus theory supply its two principal applications.
Proof and sources
Isaacs, Section 3E proves Glauberman's lemma through complement conjugacy; the complete derivation is included in the coprime-action core article. [@isaacs2008]
