Glauberman's Z* theorem for isolated involutions

glauberman_zStar

Submitter: Kim Morrison.

Notes: For a finite group G with an isolated involution t (no distinct conjugate of t commutes with t), there is a normal subgroup N ⊴ G of odd order such that every commutator g·t·g⁻¹·t⁻¹ lies in N — i.e., t is central in G/N. The hypothesis is the global form of isolation, equivalent to (but more self-contained than) the standard Sylow-local form. Proof uses modular Brauer character theory.

Source: G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966).

Informal solution: Take N = O(G), the largest normal subgroup of odd order. By Glauberman's modular character argument, isolation of t in C_G(t) implies t commutes with every element of G modulo O(G). The core of the proof is the analysis of the principal 2-block of G via Brauer characters and the Z*-style fusion theorem.

theorem declaration uses `sorry`glauberman_zStar (G : Type) [Group G] [Fintype G]
    (t : G) (ht1 : t ≠ 1) (ht2 : t * t = 1)
    (hisolated : ∀ g : G, (g * t * g⁻¹) * t = t * (g * t * g⁻¹) →
      g * t * g⁻¹ = t) :
    ∃ N : Subgroup G, N.Normal ∧ Odd (Nat.card N) ∧
      ∀ g : G, g * t * g⁻¹ * t⁻¹ ∈ N := byG:Typeinst✝¹:Group Ginst✝:Fintype Gt:Ght1:t ≠ 1ht2:t * t = 1hisolated:∀ (g : G), g * t * g⁻¹ * t = t * (g * t * g⁻¹) → g * t * g⁻¹ = t⊢ ∃ N, N.Normal ∧ Odd (Nat.card ↥N) ∧ ∀ (g : G), g * t * g⁻¹ * t⁻¹ ∈ N
  sorryAll goals completed! 🐙

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