Baer–Suzuki theorem

DefinitionBaer–Suzuki property for periodic groups at the prime 2#✓fully proved

The proposition that for every group \(G\) in which every element has finite order and every element \(a \in G\) such that any two conjugates of \(a\) generate a finite \(2\)-subgroup, there is a normal \(2\)-subgroup of \(G\) containing \(a\). Here a \(2\)-subgroup is a subgroup in which every element has order a power of \(2\).

This is the case \(p = 2\) of Kourovka Problem 11.11(a) (A. V. Borovik).

def baerSuzukiPeriodicTwo : Prop :=
  ∀ (G : Type) [Group G], (∀ g : G, IsOfFinOrder g) → ∀ a : G,
    (∀ x y : G, IsConj a x → IsConj a y →
      (Subgroup.closure {x, y} : Set G).Finite ∧ IsPGroup 2 (Subgroup.closure {x, y})) →
    ∃ N : Subgroup G, N.Normal ∧ IsPGroup 2 N ∧ a ∈ N
Used by 1
Used byFailure of the Baer–Suzuki theorem for periodic groups at the prime 2

DefinitionBaer–Suzuki property for binary finite groups#✓fully proved

The proposition that for every prime \(p\), every group \(G\) in which any two elements generate a finite subgroup, and every element \(a \in G\) such that any two conjugates of \(a\) generate a \(p\)-subgroup, there is a normal \(p\)-subgroup of \(G\) containing \(a\). Here a \(p\)-subgroup is a subgroup in which every element has order a power of \(p\).

This is the statement asked about in Kourovka Problem 11.11(b) (A. V. Borovik).

def baerSuzukiBinaryFinite : Prop :=
  ∀ p : ℕ, p.Prime → ∀ (G : Type) [Group G], (∀ x y : G, (Subgroup.closure {x, y} : Set G).Finite) →
    ∀ a : G, (∀ x y : G, IsConj a x → IsConj a y → IsPGroup p (Subgroup.closure {x, y})) →
      ∃ N : Subgroup G, N.Normal ∧ IsPGroup p N ∧ a ∈ N
Used by 1
Used byBaer–Suzuki theorem for binary finite groups

TheoremFailure of the Baer–Suzuki theorem for periodic groups at the prime 2#●ready to prove

There is a group \(G\) in which every element has finite order, together with an element \(a \in G\) such that any two conjugates of \(a\) generate a finite \(2\)-subgroup, but no normal \(2\)-subgroup of \(G\) contains \(a\). Here a \(2\)-subgroup is a subgroup in which every element has order a power of \(2\).

This answers Kourovka Problem 11.11(a) negatively for \(p = 2\) (Mazurov, Olshanskii and Sozutov, 2015).

public theorem_wanted baerSuzukiPeriodicTwo_refuted : ¬ baerSuzukiPeriodicTwo
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 11.11 (A. V. Borovik); resolution: V. D. Mazurov, A. Yu. Olshanskii and A. I. Sozutov, Algebra and Logic 54, no. 2 (2015), 161-166 (a); A. I. Sozutov, Siberian Math. J. 41, no. 3 (2000), 561-562 (b), Kourovka Archive, Problem 11.11.
Uses 1
Statement usesBaer–Suzuki property for periodic groups at the prime 2

TheoremBaer–Suzuki theorem for binary finite groups#●ready to prove

For every prime \(p\), every group \(G\) in which any two elements generate a finite subgroup, and every element \(a \in G\) such that any two conjugates of \(a\) generate a \(p\)-subgroup, there is a normal \(p\)-subgroup of \(G\) containing \(a\). Here a \(p\)-subgroup is a subgroup in which every element has order a power of \(p\).

This answers Kourovka Problem 11.11(b) affirmatively (Sozutov, 2000): by the Baer–Suzuki theorem for finite groups, \(\langle a_1, b \rangle\) is a finite \(p\)-group for every conjugate \(a_1\) of \(a\) and every element \(b\) of \(p\)-power order, so by induction every product \(a_1 \cdots a_n\) of conjugates of \(a\) has \(p\)-power order, and the normal closure of \(a\) is a normal \(p\)-subgroup.

public theorem_wanted baerSuzukiBinaryFinite_holds : baerSuzukiBinaryFinite
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 11.11 (A. V. Borovik); resolution: V. D. Mazurov, A. Yu. Olshanskii and A. I. Sozutov, Algebra and Logic 54, no. 2 (2015), 161-166 (a); A. I. Sozutov, Siberian Math. J. 41, no. 3 (2000), 561-562 (b), Kourovka Archive, Problem 11.11.
Uses 1
Statement usesBaer–Suzuki property for binary finite groups

DefinitionThe \(p\)-core of a group#✓fully proved

For a natural number \(p\) and a group \(G\), the \(p\)-core \(O_p(G)\) is the supremum, in the lattice of subgroups of \(G\), of all normal subgroups \(N\) of \(G\) that are \(p\)-groups, that is, such that every \(y \in N\) satisfies \(y^{p^k} = 1\) for some \(k \ge 0\).

When \(p\) is prime, \(O_p(G)\) is itself a normal \(p\)-subgroup of \(G\), hence the largest one, and no finiteness hypothesis is needed: the join of two normal \(p\)-subgroups is again a normal \(p\)-subgroup, so the family is directed and every element of the supremum lies in one of them.

def pCore (p : ℕ) (G : Type*) [Group G] : Subgroup G :=
  sSup {N : Subgroup G | N.Normal ∧ IsPGroup p N}
Used by 1
Used byBaer–Suzuki theorem

TheoremBaer–Suzuki theorem#●ready to prove

Let \(G\) be a finite group, \(p\) a prime and \(x \in G\). Then \(x\) lies in the \(p\)-core \(O_p(G)\) of \(G\) if and only if, for every \(g \in G\), the subgroup of \(G\) generated by \(x\) and \(g x g^{-1}\) is a \(p\)-group, that is, every element of it has order a power of \(p\).

public theorem_wanted baer_suzuki {G : Type*} [Group G] [Finite G]
    {p : ℕ} [Fact p.Prime] (x : G) :
    x ∈ pCore p G ↔
      ∀ g : G, IsPGroup p
        (Subgroup.closure ({x, g * x * g⁻¹} : Set G))
Sources
  • R. Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256-270; M. Suzuki, Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. of Math. (2) 82 (1965), 191-212. Problem baer_suzuki of the lean-eval benchmark.
Uses 1
Statement usesThe \(p\)-core of a group
3 fully proved3 ready to prove

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