Orders of automorphism groups

DefinitionDeaconescu's totient lower bound conjecture (Kourovka 15.43(a))#✓fully proved

The statement that every finite group \(G\) satisfies \(\varphi(|G|) \le |\mathrm{Aut}(G)|\), where \(\varphi\) is Euler's totient function and \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

def totientLeAutOrder : Prop :=
  ∀ (G : Type) [Group G] [Finite G], (Nat.card G).totient ≤ Nat.card (MulAut G)
Used by 1
Used byDeaconescu's totient lower bound conjecture fails (Kourovka 15.43(a))

DefinitionDeaconescu's cyclicity conjecture (Kourovka 15.43(b))#✓fully proved

The statement that every finite group \(G\) with \(|\mathrm{Aut}(G)| = \varphi(|G|)\) is cyclic, where \(\varphi\) is Euler's totient function and \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

def autOrderEqTotientImpliesCyclic : Prop :=
  ∀ (G : Type) [Group G] [Finite G], Nat.card (MulAut G) = (Nat.card G).totient → IsCyclic G
Used by 1
Used byDeaconescu's cyclicity conjecture fails (Kourovka 15.43(b))

TheoremDeaconescu's totient lower bound conjecture fails (Kourovka 15.43(a))#●ready to prove

Not every finite group \(G\) satisfies \(\varphi(|G|) \le |\mathrm{Aut}(G)|\): there is a finite group \(G\) with \(|\mathrm{Aut}(G)| < \varphi(|G|)\), where \(\varphi\) is Euler's totient function and \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

Bray and Wilson prove more than is stated here: the ratio \(|\mathrm{Aut}(G)| / \varphi(|G|)\) can be made arbitrarily small.

public theorem_wanted totientLeAutOrder_refuted : ¬ totientLeAutOrder
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 15.43 (M. Deaconescu); resolution: J. N. Bray and R. A. Wilson, On the orders of automorphism groups of finite groups, Bull. London Math. Soc. 37 (2005), no. 3, 381-385, main results (groups with |Aut G| < φ(|G|), infinitely many non-cyclic groups with |Aut G| = φ(|G|), and |Aut G|/φ(|G|) arbitrarily small); Kourovka Archive, Problem 15.43 (both answers negative), https://webspace.maths.qmul.ac.uk/j.n.bray/web/Publ/AutG/.
Uses 1
Statement usesDeaconescu's totient lower bound conjecture (Kourovka 15.43(a))

TheoremDeaconescu's cyclicity conjecture fails (Kourovka 15.43(b))#●ready to prove

Not every finite group \(G\) with \(|\mathrm{Aut}(G)| = \varphi(|G|)\) is cyclic: there is a non-cyclic finite group \(G\) with \(|\mathrm{Aut}(G)| = \varphi(|G|)\), where \(\varphi\) is Euler's totient function and \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

Bray and Wilson prove more than is stated here: there are infinitely many such non-cyclic groups.

public theorem_wanted autOrderEqTotientImpliesCyclic_refuted : ¬ autOrderEqTotientImpliesCyclic
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 15.43 (M. Deaconescu); resolution: J. N. Bray and R. A. Wilson, On the orders of automorphism groups of finite groups, Bull. London Math. Soc. 37 (2005), no. 3, 381-385, main results (groups with |Aut G| < φ(|G|), infinitely many non-cyclic groups with |Aut G| = φ(|G|), and |Aut G|/φ(|G|) arbitrarily small); Kourovka Archive, Problem 15.43 (both answers negative), https://webspace.maths.qmul.ac.uk/j.n.bray/web/Publ/AutG/.
Uses 1
Statement usesDeaconescu's cyclicity conjecture (Kourovka 15.43(b))

DefinitionPatne's prime-power divisibility problem (Kourovka 18.82)#✓fully proved

The statement that there is a function \(f : \mathbb{N} \to \mathbb{N}\) such that for every \(n \in \mathbb{N}\), every prime \(p\) and every finite group \(G\), if \(p^{f(n)}\) divides \(|G|\) then \(p^n\) divides \(|\mathrm{Aut}(G)|\), where \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

def autOrderPrimePowerBound : Prop :=
  ∃ f : ℕ → ℕ, ∀ n p : ℕ, p.Prime → ∀ (G : Type) [Group G] [Finite G],
    p ^ f n ∣ Nat.card G → p ^ n ∣ Nat.card (MulAut G)
Used by 1
Used byGreen's theorem on prime powers dividing the order of the automorphism group (Kourovka 18.82)

TheoremGreen's theorem on prime powers dividing the order of the automorphism group (Kourovka 18.82)#●ready to prove

There is a function \(f : \mathbb{N} \to \mathbb{N}\), the same for all primes, such that for every \(n \in \mathbb{N}\), every prime \(p\) and every finite group \(G\), if \(p^{f(n)}\) divides \(|G|\) then \(p^n\) divides \(|\mathrm{Aut}(G)|\), where \(\mathrm{Aut}(G)\) is the group of automorphisms of \(G\).

Ledermann and Neumann (1956) first proved this with a function depending on \(p\); J. A. Green (1956) obtained a function independent of \(p\).

public theorem_wanted autOrderPrimePowerBound_holds : autOrderPrimePowerBound
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 18.82 (R. M. Patne); resolution: J. A. Green, Proc. Roy. Soc. London Ser. A 237 (1956), 574-581 (improving W. Ledermann and B. H. Neumann, Proc. Roy. Soc. London Ser. A 235 (1956), 235-246), Kourovka Archive, Problem 18.82: Yes, there is, https://arxiv.org/abs/1401.0300v49.
Uses 1
Statement usesPatne's prime-power divisibility problem (Kourovka 18.82)

DefinitionDivisibility conjecture for orders of automorphism groups of \(p\)-groups (Kourovka 12.77)#✓fully proved

The statement that there is a prime \(p\) such that for every finite non-cyclic \(p\)-group \(P\) of order greater than \(p^2\), the order of \(P\) divides the order of its automorphism group \(\mathrm{Aut}(P)\).

The source asks this for every prime \(p\); stating it for some prime makes its refutation assert a counterexample for every prime.

def pGroupOrderDividesAutOrder : Prop :=
  ∃ p : ℕ, p.Prime ∧ ∀ (P : Type) [Group P] [Finite P], IsPGroup p P → ¬ IsCyclic P →
    p ^ 2 < Nat.card P → Nat.card P ∣ Nat.card (MulAut P)
Used by 1
Used byOrders of \(p\)-groups need not divide orders of their automorphism groups

TheoremOrders of \(p\)-groups need not divide orders of their automorphism groups#●ready to prove

For every prime \(p\) there is a finite non-cyclic \(p\)-group \(P\) of order greater than \(p^2\) whose order does not divide the order of its automorphism group \(\mathrm{Aut}(P)\).

González-Sánchez and Jaikin-Zapirain construct, for each prime \(p\), finite \(p\)-groups \(G_i\) with \(|\mathrm{Aut}(G_i)| / |G_i| \to 0\); they are non-abelian, hence non-cyclic of order greater than \(p^2\), and eventually \(|\mathrm{Aut}(G_i)| < |G_i|\).

public theorem_wanted pGroupOrderDividesAutOrder_refuted : ¬ pGroupOrderDividesAutOrder
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 12.77 (well-known problem; A. I. Starostin); resolution: J. González-Sánchez and A. Jaikin-Zapirain, Finite p-groups with small automorphism group, Forum Math. Sigma 3 (2015), Paper No. e7, main theorem (abstract): for each prime p, finite p-groups G_i with |Aut(G_i)|/|G_i| → 0; Kourovka Archive, Problem 12.77.
Uses 1
Statement usesDivisibility conjecture for orders of automorphism groups of \(p\)-groups (Kourovka 12.77)
4 fully proved4 ready to prove

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