FormalConjectures.Kourovka.«8

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import FormalConjecturesUtil

Conjecture 8.8

by R. I. Grigorchuk

Both parts of the problem are due to D. V. Anosov.

Reference: The Kourovka Notebook

namespace Kourovka.«8.8»

Does there exist a non-cyclic finitely-generated group $G$ containing an element $a$ such that every element of $G$ is conjugate to a power of $a$?

Here a power of $a$ means $a^n$ for some $n \in \mathbb{Z}$.

Answered affirmatively by V. S. Guba, Math. USSR-Izv. 29 (1986), 233–277.

@[category research solved, AMS 20]
theorem kourovka_8_8a : answer(True) ↔
    ∃ (G : Type) (_ : Group G), ¬ IsCyclic G ∧ Group.FG G ∧
      ∃ a : G, ∀ g : G, ∃ n : ℤ, IsConj g (a ^ n) := by⊢ True ↔ ∃ G x, ¬IsCyclic G ∧ Group.FG G ∧ ∃ a, ∀ (g : G), ∃ n, IsConj g (a ^ n)
  sorryAll goals completed! 🐙

Does there exist a non-cyclic finitely presented group $G$ which contains an element $a$ such that each element of $G$ is conjugate to some power of $a$?

Here a power of $a$ means $a^n$ for some $n \in \mathbb{Z}$.

@[category research open, AMS 20]
theorem kourovka_8_8b : answer(sorry) ↔
    ∃ (G : Type) (_ : Group G), ¬ IsCyclic G ∧ Group.IsFinitelyPresented G ∧
      ∃ a : G, ∀ g : G, ∃ n : ℤ, IsConj g (a ^ n) := by⊢ True ↔ ∃ G x, ¬IsCyclic G ∧ Group.IsFinitelyPresented G ∧ ∃ a, ∀ (g : G), ∃ n, IsConj g (a ^ n)
  sorryAll goals completed! 🐙

end Kourovka.«8.8»