Intersections of conjugate nilpotent subgroups

DefinitionKourovka Problem 15.40#✓fully proved

The assertion, in the two-subgroup form of its resolution, that for every finite non-abelian simple group \(G\) and all nilpotent subgroups \(A\) and \(B\) of \(G\) there is \(x \in G\) with \(A \cap x B x^{-1} = 1\).

The case \(A = B\) is the problem as posed.

def nilpotentSubgroupsTrivialConjugateIntersection : Prop :=
  ∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
    ∀ A B : Subgroup G, Group.IsNilpotent A → Group.IsNilpotent B →
      ∃ x : G, A ⊓ B.map (MulAut.conj x).toMonoidHom = ⊥

Used by 1

Used byNilpotent subgroups of a finite simple group have conjugates meeting trivially

TheoremNilpotent subgroups of a finite simple group have conjugates meeting trivially#●ready to prove

Let \(G\) be a finite non-abelian simple group and let \(A\) and \(B\) be nilpotent subgroups of \(G\). Then there is \(x \in G\) with \(A \cap x B x^{-1} = 1\).

Burness and Huang proved this (for alternating groups using work of Kurmazov and Zenkov); the case \(A = B\) answers Kourovka Problem 15.40 positively.

public theorem_wanted nilpotentSubgroupsTrivialConjugateIntersection_holds :
    nilpotentSubgroupsTrivialConjugateIntersection

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 15.40 (E. P. Vdovin); resolution: T. C. Burness and H. Y. Huang, On the intersections of nilpotent subgroups in simple groups, arXiv:2508.03479 (2025), abstract (Vdovin's conjecture and its stronger form: A ∩ B^x = 1 for nilpotent subgroups A, B of a simple group, combined with work of Kurmazov and Zenkov on alternating groups); the Kourovka Notebook (21st issue) marks 15.40 solved, https://arxiv.org/abs/2508.03479.

Uses 1

Statement usesKourovka Problem 15.40

DefinitionFitting subgroup#✓fully proved

The Fitting subgroup F(G): the join of the nilpotent normal subgroups of G (for finite G the largest nilpotent normal subgroup).

def fittingSubgroup (G : Type*) [Group G] : Subgroup G :=
  ⨆ (N : Subgroup G) (_ : N.Normal) (_ : Group.IsNilpotent N), N

Used by 5

Used byInclusion-minimal triple intersections and the Fitting subgroup (Kourovka 20.122(b), (c)) · Minimal-order triple intersections and the Fitting subgroup (Kourovka 20.122(a), (c)) · Kourovka Problem 19.37(a) · Kourovka Problem 19.37(b) · Kourovka Problem 17.40

DefinitionKourovka Problem 17.40#✓fully proved

The assertion that for every finite group \(G\) and every nilpotent subgroup \(N\) of \(G\) there are \(x, y \in G\) with \(N \cap x N x^{-1} \cap y N y^{-1} \le F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

def nilpotentThreeConjugatesInFitting : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (N : Subgroup G), Group.IsNilpotent N →
    ∃ x y : G, N ⊓ N.map (MulAut.conj x).toMonoidHom ⊓ N.map (MulAut.conj y).toMonoidHom ≤
      fittingSubgroup G

Uses 1 · Used by 1

Statement usesFitting subgroup

Used byThree conjugates of a nilpotent subgroup meet inside the Fitting subgroup

TheoremThree conjugates of a nilpotent subgroup meet inside the Fitting subgroup#●ready to prove

For every finite group \(G\) and every nilpotent subgroup \(N\) of \(G\) there are \(x, y \in G\) with \(N \cap x N x^{-1} \cap y N y^{-1} \le F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

This answers Kourovka Problem 17.40 positively; it is due to Zenkov (2021) and uses the classification of finite simple groups.

public theorem_wanted nilpotentThreeConjugatesInFitting_holds : nilpotentThreeConjugatesInFitting

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 17.40 (E. P. Vdovin); resolution: V. I. Zenkov, Siberian Math. J. 62 (2021), 621-637, Kourovka Archive, Problem 17.40 (Yes, such elements always exist, mod CFSG).

Uses 1

Statement usesKourovka Problem 17.40

DefinitionTriple intersections with conjugates#✓fully proved

For subgroups \(A\), \(B\), \(C\) of a group \(G\), this is the set of all subgroups of \(G\) of the form \(A \cap xBx^{-1} \cap yCy^{-1}\) with \(x, y \in G\), that is, the intersections of \(A\) with a conjugate of \(B\) and a conjugate of \(C\).

def tripleIntersections {G : Type} [Group G] (A B C : Subgroup G) : Set (Subgroup G) :=
  {D | ∃ x y : G,
    D = A ⊓ B.map (MulAut.conj x).toMonoidHom ⊓ C.map (MulAut.conj y).toMonoidHom}

Used by 2

Used bySubgroup generated by the inclusion-minimal triple intersections · Subgroup generated by the triple intersections of least order

DefinitionSubgroup generated by the inclusion-minimal triple intersections#✓fully proved

For subgroups \(A\), \(B\), \(C\) of a group \(G\), the subgroup \(\mathrm{Min}_G(A, B, C)\) is the subgroup of \(G\) generated by those intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) that are minimal by inclusion among all such intersections, that is, that contain no other intersection of this form.

def inclusionMinimalIntersectionJoin {G : Type} [Group G] (A B C : Subgroup G) : Subgroup G :=
  sSup {D | D ∈ tripleIntersections A B C ∧ ∀ E ∈ tripleIntersections A B C, E ≤ D → E = D}

Uses 1 · Used by 1

Statement usesTriple intersections with conjugates

Used byInclusion-minimal triple intersections and the Fitting subgroup (Kourovka 20.122(b), (c))

DefinitionSubgroup generated by the triple intersections of least order#✓fully proved

For subgroups \(A\), \(B\), \(C\) of a group \(G\), the subgroup \(\min_G(A, B, C)\) is the subgroup of \(G\) generated by those intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) whose order is at most the order of every such intersection.

Orders are taken as natural numbers, so an infinite subgroup counts as having order \(0\); the notion is intended for finite \(G\).

def minimalOrderIntersectionJoin {G : Type} [Group G] (A B C : Subgroup G) : Subgroup G :=
  sSup {D | D ∈ tripleIntersections A B C ∧
    ∀ E ∈ tripleIntersections A B C, Nat.card D ≤ Nat.card E}

Uses 1 · Used by 1

Statement usesTriple intersections with conjugates

Used byMinimal-order triple intersections and the Fitting subgroup (Kourovka 20.122(a), (c))

DefinitionMinimal-order triple intersections and the Fitting subgroup (Kourovka 20.122(a), (c))#✓fully proved

The statement that for every finite soluble group \(G\) and all nilpotent subgroups \(A\), \(B\), \(C\) of \(G\), the subgroup \(\min_G(A, B, C)\) generated by the intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) of least order is contained in the Fitting subgroup \(F(G)\), the join of all nilpotent normal subgroups of \(G\).

The question is due to V. I. Zenkov.

def minimalOrderIntersectionsInFitting : Prop :=
  ∀ (G : Type) [Group G] [Finite G], Group.IsSolvable G → ∀ A B C : Subgroup G,
    Group.IsNilpotent A → Group.IsNilpotent B → Group.IsNilpotent C →
      minimalOrderIntersectionJoin A B C ≤ fittingSubgroup G

Uses 2 · Used by 1

Statement usesSubgroup generated by the triple intersections of least order · Fitting subgroup

Used byMinimal-order triple intersections need not lie in the Fitting subgroup

DefinitionInclusion-minimal triple intersections and the Fitting subgroup (Kourovka 20.122(b), (c))#✓fully proved

The statement that for every finite soluble group \(G\) and all nilpotent subgroups \(A\), \(B\), \(C\) of \(G\), the subgroup \(\mathrm{Min}_G(A, B, C)\) generated by those intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) that are minimal by inclusion among all such intersections is contained in the Fitting subgroup \(F(G)\), the join of all nilpotent normal subgroups of \(G\).

def inclusionMinimalIntersectionsInFitting : Prop :=
  ∀ (G : Type) [Group G] [Finite G], Group.IsSolvable G → ∀ A B C : Subgroup G,
    Group.IsNilpotent A → Group.IsNilpotent B → Group.IsNilpotent C →
      inclusionMinimalIntersectionJoin A B C ≤ fittingSubgroup G

Uses 2 · Used by 1

Statement usesSubgroup generated by the inclusion-minimal triple intersections · Fitting subgroup

Used byInclusion-minimal triple intersections need not lie in the Fitting subgroup

TheoremMinimal-order triple intersections need not lie in the Fitting subgroup#●ready to prove

It is not true that for every finite soluble group \(G\) and all nilpotent subgroups \(A\), \(B\), \(C\) of \(G\), the subgroup \(\min_G(A, B, C)\) generated by the intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) of least order is contained in the Fitting subgroup \(F(G)\), the join of all nilpotent normal subgroups of \(G\).

Kinyon and Urban (Section 3.38) give a soluble group \(G\) of order \(36864\) with nilpotent subgroups \(A\) of order \(16\) and \(B = C\) of order \(2048\) such that \(\min_G(A, B, B) = A\) while \(F(G)\) does not contain \(A\). Since this group is soluble, it answers part (c), and hence part (a), in the negative.

public theorem_wanted minimalOrderIntersectionsInFitting_refuted :
    ¬ minimalOrderIntersectionsInFitting

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.122 (V. I. Zenkov); resolution: M. Kinyon and J. Urban, Results on problems from the Kourovka Notebook, preprint, version 4, 19 September 2026, Kinyon-Urban, Section 3.38 (Problem 20.122(a)-(c)): G = E ⋊ Q soluble of order 512 · 72 = 36864, A = Z × H of order 16 and B = C the stabilizer of order 2048 are nilpotent, min_G(A, B, B) = Min_G(A, B, B) = A and F(G) = E does not contain A; the Kourovka Notebook (21st issue) marks 20.122 solved, "No, it is not always true in all three cases" (AI solution confirmed by V. I. Zenkov), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesMinimal-order triple intersections and the Fitting subgroup (Kourovka 20.122(a), (c))

TheoremInclusion-minimal triple intersections need not lie in the Fitting subgroup#●ready to prove

It is not true that for every finite soluble group \(G\) and all nilpotent subgroups \(A\), \(B\), \(C\) of \(G\), the subgroup \(\mathrm{Min}_G(A, B, C)\) generated by those intersections \(A \cap xBx^{-1} \cap yCy^{-1}\) (\(x, y \in G\)) that are minimal by inclusion among all such intersections is contained in the Fitting subgroup \(F(G)\), the join of all nilpotent normal subgroups of \(G\).

Kinyon and Urban (Section 3.38) give a soluble group \(G\) of order \(36864\) with nilpotent subgroups \(A\) of order \(16\) and \(B = C\) of order \(2048\) such that \(\mathrm{Min}_G(A, B, B) = A\) while \(F(G)\) does not contain \(A\). Since this group is soluble, it answers part (c), and hence part (b), in the negative.

public theorem_wanted inclusionMinimalIntersectionsInFitting_refuted :
    ¬ inclusionMinimalIntersectionsInFitting

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.122 (V. I. Zenkov); resolution: M. Kinyon and J. Urban, Results on problems from the Kourovka Notebook, preprint, version 4, 19 September 2026, Kinyon-Urban, Section 3.38 (Problem 20.122(a)-(c)): G = E ⋊ Q soluble of order 512 · 72 = 36864, A = Z × H of order 16 and B = C the stabilizer of order 2048 are nilpotent, min_G(A, B, B) = Min_G(A, B, B) = A and F(G) = E does not contain A; the Kourovka Notebook (21st issue) marks 20.122 solved, "No, it is not always true in all three cases" (AI solution confirmed by V. I. Zenkov), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesInclusion-minimal triple intersections and the Fitting subgroup (Kourovka 20.122(b), (c))

DefinitionNilpotent injector#✓fully proved

A subgroup \(H\) of a group \(G\) is a nilpotent injector when, for every subnormal subgroup \(K\) of \(G\), the intersection \(H \cap K\) is nilpotent and is a maximal nilpotent subgroup of \(K\): every nilpotent subgroup \(L\) of \(G\) with \(H \cap K \le L \le K\) equals \(H \cap K\).

def IsNilpotentInjector {G : Type*} [Group G] (H : Subgroup G) : Prop :=
  ∀ K : Subgroup G, K.IsSubnormal → Group.IsNilpotent ↥(H ⊓ K) ∧
    ∀ L : Subgroup G, L ≤ K → Group.IsNilpotent L → H ⊓ K ≤ L → L = H ⊓ K

Used by 2

Used byKourovka Problem 19.37(a) · Kourovka Problem 19.37(b)

DefinitionKourovka Problem 19.37(a)#✓fully proved

The assertion that there is a natural number \(k\) such that for every finite group \(G\) and every nilpotent injector \(H\) of \(G\) there are elements \(g_1, \dots, g_k \in G\), not necessarily distinct, with \(g_1 H g_1^{-1} \cap \dots \cap g_k H g_k^{-1} = F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

def nilpotentInjectorBoundedConjugates : Prop :=
  ∃ k : ℕ, ∀ (G : Type) [Group G] [Finite G] (H : Subgroup G), IsNilpotentInjector H →
    ∃ g : Fin k → G, ⨅ i, H.map (MulAut.conj (g i)).toMonoidHom = fittingSubgroup G

Uses 2 · Used by 1

Statement usesNilpotent injector · Fitting subgroup

Used byBoundedly many conjugates of a nilpotent injector meet in the Fitting subgroup

DefinitionKourovka Problem 19.37(b)#✓fully proved

The assertion that for every finite group \(G\) and every nilpotent injector \(H\) of \(G\) there are elements \(g_1, g_2, g_3 \in G\), not necessarily distinct, with \(g_1 H g_1^{-1} \cap g_2 H g_2^{-1} \cap g_3 H g_3^{-1} = F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

def nilpotentInjectorThreeConjugates : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (H : Subgroup G), IsNilpotentInjector H →
    ∃ g : Fin 3 → G, ⨅ i, H.map (MulAut.conj (g i)).toMonoidHom = fittingSubgroup G

Uses 2 · Used by 1

Statement usesNilpotent injector · Fitting subgroup

Used byThree conjugates of a nilpotent injector meet in the Fitting subgroup

TheoremBoundedly many conjugates of a nilpotent injector meet in the Fitting subgroup#●ready to prove

There is a natural number \(k\) such that for every finite group \(G\) and every nilpotent injector \(H\) of \(G\) there are elements \(g_1, \dots, g_k \in G\), not necessarily distinct, with \(g_1 H g_1^{-1} \cap \dots \cap g_k H g_k^{-1} = F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

Zenkov proved, using the classification of finite simple groups, that \(k = 3\) works, which answers Kourovka Problem 19.37(a) positively.

public theorem_wanted nilpotentInjectorBoundedConjugates_holds : nilpotentInjectorBoundedConjugates

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 19.37 (S. F. Kamornikov); resolution: V. I. Zenkov, Intersections of three nilpotent subgroups in a finite group, Siberian Math. J. 62 (2021), no. 4, 621-637, doi:10.1134/S0037446621040066, Kourovka Archive, Problem 19.37: Yes, one can choose k = 3 as such a constant, mod CFSG, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 19.37(a)

TheoremThree conjugates of a nilpotent injector meet in the Fitting subgroup#●ready to prove

For every finite group \(G\) and every nilpotent injector \(H\) of \(G\) there are elements \(g_1, g_2, g_3 \in G\), not necessarily distinct, with \(g_1 H g_1^{-1} \cap g_2 H g_2^{-1} \cap g_3 H g_3^{-1} = F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\).

This answers Kourovka Problem 19.37(b) positively; it is due to Zenkov and uses the classification of finite simple groups, the soluble case being due to Passman and Mann.

public theorem_wanted nilpotentInjectorThreeConjugates_holds : nilpotentInjectorThreeConjugates

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 19.37 (S. F. Kamornikov); resolution: V. I. Zenkov, Intersections of three nilpotent subgroups in a finite group, Siberian Math. J. 62 (2021), no. 4, 621-637, doi:10.1134/S0037446621040066, Kourovka Archive, Problem 19.37: Yes, one can choose k = 3 as such a constant, mod CFSG, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 19.37(b)

11 fully proved6 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.