Squares of 2-automorphisms

DefinitionKourovka Problem 14.1#✓fully proved

The statement that for every finite group \(G\) with no nontrivial normal subgroup of odd order and every automorphism \(\varphi\) of \(G\) whose order is a power of \(2\), if \(\varphi\) fixes every element of some Sylow \(2\)-subgroup of \(G\), then \(\varphi^2\) is conjugation by some element of \(G\), that is, \(\varphi^2\) is an inner automorphism.

def twoAutomorphismCentralizingSylowSquareInner : Prop :=
  ∀ (G : Type) [Group G] [Finite G],
    (∀ N : Subgroup G, N.Normal → Odd (Nat.card N) → N = ⊥) →
    ∀ φ : MulAut G, (∃ k : ℕ, orderOf φ = 2 ^ k) →
      (∃ S : Sylow 2 G, ∀ s ∈ (S : Subgroup G), φ s = s) → ∃ g : G, φ ^ 2 = MulAut.conj g

Used by 1

Used byGlauberman's theorem on squares of \(2\)-automorphisms (Kourovka Problem 14.1)

TheoremGlauberman's theorem on squares of \(2\)-automorphisms (Kourovka Problem 14.1)#●ready to prove

Let \(G\) be a finite group with no nontrivial normal subgroup of odd order, and let \(\varphi\) be an automorphism of \(G\) whose order is a power of \(2\). If \(\varphi\) fixes every element of some Sylow \(2\)-subgroup of \(G\), then \(\varphi^2\) is an inner automorphism of \(G\), that is, \(\varphi^2\) is conjugation by some \(g \in G\).

public theorem_wanted twoAutomorphismCentralizingSylowSquareInner_holds :
    twoAutomorphismCentralizingSylowSquareInner

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 14.1 (R. Zh. Aleev); resolution: G. Glauberman, Math. Z. 107 (1968), 1-20, Kourovka Archive, Problem 14.1 (answer: Yes, it is true).

Uses 1

Statement usesKourovka Problem 14.1

1 fully proved1 ready to prove

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