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Finite groups with a metacyclic Frobenius group of automorphisms - - PowerPoint PPT Presentation

Oct 16, 2023 20 likes •691 views

Finite groups with a metacyclic Frobenius group of automorphisms Natalia Makarenko (main results are joint with E. I. Khukhro) Universit de Haute Alsace, Sobolev Institute of Mathematics, Novosibirsk Groups St Andrews 2013 N. Makarenko

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Finite groups with a metacyclic Frobenius group of automorphisms

Natalia Makarenko (main results are joint with E. I. Khukhro)

Université de Haute Alsace, Sobolev Institute of Mathematics, Novosibirsk

Groups St Andrews 2013

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties H acts faithfully on every non-trivial H-invariant subgroup of F. It follows that all abelian subgroups of H are cyclic.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties H acts faithfully on every non-trivial H-invariant subgroup of F. It follows that all abelian subgroups of H are cyclic. If F is cyclic, then H is cyclic.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties H acts faithfully on every non-trivial H-invariant subgroup of F. It follows that all abelian subgroups of H are cyclic. If F is cyclic, then H is cyclic. By Thompson’s theorem (1959) a finite group admitting a fixed-point-free automorphism of prime order is nilpotent. Hence F is nilpotent.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties H acts faithfully on every non-trivial H-invariant subgroup of F. It follows that all abelian subgroups of H are cyclic. If F is cyclic, then H is cyclic. By Thompson’s theorem (1959) a finite group admitting a fixed-point-free automorphism of prime order is nilpotent. Hence F is nilpotent. By Higman’s theorem (1957) the nilpotency class of a finite (nilpotent) group admitting a fixed-point-free automorphism of prime order p is bounded in terms of p. Hence the nilpotency class of F is bounded in terms of the least prime divisor of |H|.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Frobenius group

  • Definition. A Frobenius group FH with kernel F and complement H is a

semidirect product of a normal (finite) subgroup F on which H acts by automorphisms so that CF(h) = 1 for every h ∈ H \ {1}. Properties H acts faithfully on every non-trivial H-invariant subgroup of F. It follows that all abelian subgroups of H are cyclic. If F is cyclic, then H is cyclic. By Thompson’s theorem (1959) a finite group admitting a fixed-point-free automorphism of prime order is nilpotent. Hence F is nilpotent. By Higman’s theorem (1957) the nilpotency class of a finite (nilpotent) group admitting a fixed-point-free automorphism of prime order p is bounded in terms of p. Hence the nilpotency class of F is bounded in terms of the least prime divisor of |H|. An explicit upper bound for nilpotency class is given by Kreknin

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

  • Definition. If a Frobenius group FH acts on a group G in such a

manner that GF is also a Frobenius group, then the product GFH is called a double Frobenius group.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

  • Definition. If a Frobenius group FH acts on a group G in such a

manner that GF is also a Frobenius group, then the product GFH is called a double Frobenius group. Theorem (Mazurov, 2002) If GFH is a double Frobenius group such that CG(H) is abelian and H is of order 2 or 3, then G is nilpotent of class at most 2.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

  • Definition. If a Frobenius group FH acts on a group G in such a

manner that GF is also a Frobenius group, then the product GFH is called a double Frobenius group. Theorem (Mazurov, 2002) If GFH is a double Frobenius group such that CG(H) is abelian and H is of order 2 or 3, then G is nilpotent of class at most 2. Mazurov’s Problem (Kourovka Notebook, Problem 17.72). Let GFH be a double Frobenius group.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

  • Definition. If a Frobenius group FH acts on a group G in such a

manner that GF is also a Frobenius group, then the product GFH is called a double Frobenius group. Theorem (Mazurov, 2002) If GFH is a double Frobenius group such that CG(H) is abelian and H is of order 2 or 3, then G is nilpotent of class at most 2. Mazurov’s Problem (Kourovka Notebook, Problem 17.72). Let GFH be a double Frobenius group. Part (a). Can the nilpotency class of G be bounded in terms of |H| and the nilpotency class of CG(H)?

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

  • Definition. If a Frobenius group FH acts on a group G in such a

manner that GF is also a Frobenius group, then the product GFH is called a double Frobenius group. Theorem (Mazurov, 2002) If GFH is a double Frobenius group such that CG(H) is abelian and H is of order 2 or 3, then G is nilpotent of class at most 2. Mazurov’s Problem (Kourovka Notebook, Problem 17.72). Let GFH be a double Frobenius group. Part (a). Can the nilpotency class of G be bounded in terms of |H| and the nilpotency class of CG(H)? Part (b). Can the exponent of G be bounded in terms of |H| and the exponent of CG(H)?

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

Theorem (Khukhro, 2008) Let GFH be a double Frobenius group with complement H of order q. Assume that CG(H) is abelian. Then G is nilpotent of q-bounded class.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Double Frobenius groups

Theorem (Khukhro, 2008) Let GFH be a double Frobenius group with complement H of order q. Assume that CG(H) is abelian. Then G is nilpotent of q-bounded class. Theorem (Makarenko–Shumyatsky, 2010, Proc. AMS) Let GFH be a double Frobenius group with complement H of order q. Assume that CG(H) is nilpotent of class c. Then G is nilpotent of (c, q)-bounded class. Positive answer to the part a) of Mazurov’s question.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Fixed-point-free action of Frobenius kernel

Let G be a finite group. A Frobenius group FH with kernel F acts on G by automorphisms. Assume that CG(F) = 1.

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Frobenius groups of automorphisms

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Fixed-point-free action of Frobenius kernel

Let G be a finite group. A Frobenius group FH with kernel F acts on G by automorphisms. Assume that CG(F) = 1. Theorem (Makarenko–Shumyatsky, 2010, Proc. AMS) If F is cyclic of prime order, CG(F) = 1 and CG(H) is nilpotent of class c, then the nilpotency class of G is bounded in terms of |H| and c.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Fixed-point-free action of Frobenius kernel

Let G be a finite group. A Frobenius group FH with kernel F acts on G by automorphisms. Assume that CG(F) = 1. Theorem (Makarenko–Shumyatsky, 2010, Proc. AMS) If F is cyclic of prime order, CG(F) = 1 and CG(H) is nilpotent of class c, then the nilpotency class of G is bounded in terms of |H| and c.

  • Remark. By Higman-Kreknin-Kostrikin Theorem G is nilpotent of class

at most h(p), where h(p) is Higman’s function.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Fixed-point-free action of Frobenius kernel

Let G be a finite group. A Frobenius group FH with kernel F acts on G by automorphisms. Assume that CG(F) = 1. Theorem (Makarenko–Shumyatsky, 2010, Proc. AMS) If F is cyclic of prime order, CG(F) = 1 and CG(H) is nilpotent of class c, then the nilpotency class of G is bounded in terms of |H| and c.

  • Remark. By Higman-Kreknin-Kostrikin Theorem G is nilpotent of class

at most h(p), where h(p) is Higman’s function.

  • Hypothesis. Principal properties (nilpotency class, exponent, order,

rank) of G are completely defined by H and the action of H on G (and do not depend on F).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Order, rank, nilpotency

Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum) Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H and CG(F) = 1. Then the following statements hold.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Order, rank, nilpotency

Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum) Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H and CG(F) = 1. Then the following statements hold. |G| = |CG(H)||H|.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Order, rank, nilpotency

Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum) Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H and CG(F) = 1. Then the following statements hold. |G| = |CG(H)||H|. The rank of G is bounded in terms of |H| and the rank of CG(H).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Order, rank, nilpotency

Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum) Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H and CG(F) = 1. Then the following statements hold. |G| = |CG(H)||H|. The rank of G is bounded in terms of |H| and the rank of CG(H). If CG(H) is nilpotent, then G is also nilpotent.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Order, rank, nilpotency

Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum) Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H and CG(F) = 1. Then the following statements hold. |G| = |CG(H)||H|. The rank of G is bounded in terms of |H| and the rank of CG(H). If CG(H) is nilpotent, then G is also nilpotent. The proof is based on the fact that all FH-invariant elementary abelian sections of G are free FpH-modules (for various p) by Clifford’s theorem.

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Frobenius groups of automorphisms

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  • Definition. Recall that for a group A and a field k, a free kA-module of

rank n is a direct sum of n copies of the group algebra kA each of which is regarded as a vector space over k of dimension |A| with a basis {vg, g ∈ A} labelled by elements of A on which A acts in a regular permutation representation: vgh = vgh.

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Frobenius groups of automorphisms

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Exponent

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Frobenius groups of automorphisms

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Exponent

Let FH be a Frobenius group with kernel F and complement H. Suppose that CG(F) = 1 and F is cyclic (then H is also cyclic as Frobenius complement).

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Frobenius groups of automorphisms

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Exponent

Let FH be a Frobenius group with kernel F and complement H. Suppose that CG(F) = 1 and F is cyclic (then H is also cyclic as Frobenius complement). Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum). The exponent of G is bounded in terms of |FH| and the exponent of CG(H).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Exponent

Let FH be a Frobenius group with kernel F and complement H. Suppose that CG(F) = 1 and F is cyclic (then H is also cyclic as Frobenius complement). Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum). The exponent of G is bounded in terms of |FH| and the exponent of CG(H). Proof uses

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Exponent

Let FH be a Frobenius group with kernel F and complement H. Suppose that CG(F) = 1 and F is cyclic (then H is also cyclic as Frobenius complement). Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum). The exponent of G is bounded in terms of |FH| and the exponent of CG(H). Proof uses Lazard’s Lie algebra associated with the Jennings–Zassenhaus filtration;

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Exponent

Let FH be a Frobenius group with kernel F and complement H. Suppose that CG(F) = 1 and F is cyclic (then H is also cyclic as Frobenius complement). Theorem (Khukhro–Makarenko–Shumyatsky, 2010, to appear in Forum Mathematicum). The exponent of G is bounded in terms of |FH| and the exponent of CG(H). Proof uses Lazard’s Lie algebra associated with the Jennings–Zassenhaus filtration; powerful subgroups.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class

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Frobenius groups of automorphisms

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Nilpotency class

Theorem (Khu–Ma–Shu, 2010, to appear in Forum Mathematicum). Let a finite group G admit a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H of order q. Suppose that CG(F) = 1 and CG(H) is nilpotent of class c .

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class

Theorem (Khu–Ma–Shu, 2010, to appear in Forum Mathematicum). Let a finite group G admit a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H of order q. Suppose that CG(F) = 1 and CG(H) is nilpotent of class c . Then the nilpotency class of G is bounded in terms of q and c.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class

Theorem (Khu–Ma–Shu, 2010, to appear in Forum Mathematicum). Let a finite group G admit a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H of order q. Suppose that CG(F) = 1 and CG(H) is nilpotent of class c . Then the nilpotency class of G is bounded in terms of q and c. Lie ring Theorem. Let a Lie ring L admit a Frobenius group of automorphisms FH with cyclic kernel F of finite order n and complement H of order q.

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Frobenius groups of automorphisms

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Nilpotency class

Theorem (Khu–Ma–Shu, 2010, to appear in Forum Mathematicum). Let a finite group G admit a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H of order q. Suppose that CG(F) = 1 and CG(H) is nilpotent of class c . Then the nilpotency class of G is bounded in terms of q and c. Lie ring Theorem. Let a Lie ring L admit a Frobenius group of automorphisms FH with cyclic kernel F of finite order n and complement H of order q. If CL(F) = 0 and CL(H) is nilpotent of class c, then the nilpotency class of L is bounded in terms of q and c.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class

Theorem (Khu–Ma–Shu, 2010, to appear in Forum Mathematicum). Let a finite group G admit a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H of order q. Suppose that CG(F) = 1 and CG(H) is nilpotent of class c . Then the nilpotency class of G is bounded in terms of q and c. Lie ring Theorem. Let a Lie ring L admit a Frobenius group of automorphisms FH with cyclic kernel F of finite order n and complement H of order q. If CL(F) = 0 and CL(H) is nilpotent of class c, then the nilpotency class of L is bounded in terms of q and c. Note, that by Kreknin’s theorem L is solvable of n-bounded derived length.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Almost fixed-point-free action of Frobenius kernel

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Almost fixed-point-free action of Frobenius kernel

Let a Frobenius group FH with kernel F and complement H act on G by automorphisms and |CG(F)| = m.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Almost fixed-point-free action of Frobenius kernel

Let a Frobenius group FH with kernel F and complement H act on G by automorphisms and |CG(F)| = m. The aim: to obtain similar restrictions, as in the case of fixed-point-free action, for a subgroup of index bounded in terms of |CG(F)| and other parameters.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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10 / 17

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Almost fixed-point-free action of Frobenius kernel

Let a Frobenius group FH with kernel F and complement H act on G by automorphisms and |CG(F)| = m. The aim: to obtain similar restrictions, as in the case of fixed-point-free action, for a subgroup of index bounded in terms of |CG(F)| and other parameters. Khukhro (2013, Algebra and Logic): Restrictions for the order and rank

  • f G.
  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class, main results

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Frobenius groups of automorphisms

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Nilpotency class, main results

Main Theorem (Khukhro, Makarenko, J. of Algebra 2013). Let a finite group G admit a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H of order q. Suppose that the CG(H) is nilpotent of class c.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class, main results

Main Theorem (Khukhro, Makarenko, J. of Algebra 2013). Let a finite group G admit a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H of order q. Suppose that the CG(H) is nilpotent of class c. Then G has a nilpotent characteristic subgroup of index bounded in terms of c, |CG(F)|, and |FH| whose nilpotency class is bounded in terms of c and |H| only.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class, main results

Main Theorem (Khukhro, Makarenko, J. of Algebra 2013). Let a finite group G admit a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H of order q. Suppose that the CG(H) is nilpotent of class c. Then G has a nilpotent characteristic subgroup of index bounded in terms of c, |CG(F)|, and |FH| whose nilpotency class is bounded in terms of c and |H| only. Proof.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class, main results

Main Theorem (Khukhro, Makarenko, J. of Algebra 2013). Let a finite group G admit a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H of order q. Suppose that the CG(H) is nilpotent of class c. Then G has a nilpotent characteristic subgroup of index bounded in terms of c, |CG(F)|, and |FH| whose nilpotency class is bounded in terms of c and |H| only. Proof. Reduction to soluble groups is given by results based on the classification of Hartley (1992) and Wang–Cheng (1993).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency class, main results

Main Theorem (Khukhro, Makarenko, J. of Algebra 2013). Let a finite group G admit a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H of order q. Suppose that the CG(H) is nilpotent of class c. Then G has a nilpotent characteristic subgroup of index bounded in terms of c, |CG(F)|, and |FH| whose nilpotency class is bounded in terms of c and |H| only. Proof. Reduction to soluble groups is given by results based on the classification of Hartley (1992) and Wang–Cheng (1993). Reduction to the case of a nilpotent group G by bounding the index of the Fitting subgroup by representation theory arguments.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency

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Frobenius groups of automorphisms

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Nilpotency

Theorem (Khukhro, Makarenko, J. of Algebra 2013). Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such that the fixed-point subgroup CG(H) of the complement is nilpotent.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Nilpotency

Theorem (Khukhro, Makarenko, J. of Algebra 2013). Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such that the fixed-point subgroup CG(H) of the complement is nilpotent. Then the index of the Fitting subgroup F(G) is bounded in terms of |CG(F)| and |F|.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

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Proof of the main theorem

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Frobenius groups of automorphisms

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Proof of the main theorem

Proof is based on a similar Lie ring theorem.

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Frobenius groups of automorphisms

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Proof of the main theorem

Proof is based on a similar Lie ring theorem. Its application to the case of nilpotent group is not straightforward and requires additional efforts.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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SLIDE 54

Proof of the main theorem

Proof is based on a similar Lie ring theorem. Its application to the case of nilpotent group is not straightforward and requires additional efforts. Direct application of Lie ring Theorem to the associated Lie ring L(G) does not give a required result for the group G, since there is no good correspondence between subgroups of G and subrings of L(G).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

13 / 17

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SLIDE 55

Proof of the main theorem

Proof is based on a similar Lie ring theorem. Its application to the case of nilpotent group is not straightforward and requires additional efforts. Direct application of Lie ring Theorem to the associated Lie ring L(G) does not give a required result for the group G, since there is no good correspondence between subgroups of G and subrings of L(G). We overcome this difficulty by proving that, in a certain critical situation, the group G itself is nilpotent of (c, q)-bounded class, the advantage being that the nilpotency class of G is equal to that of L(G). This is not true in general, but can be achieved by using induction on a certain complex parameter.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

13 / 17

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SLIDE 56

Lie ring theorem

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

14 / 17

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SLIDE 57

Lie ring theorem

Lie ring Theorem (Khukhro, Makarenko, J. of Algebra 2013) Suppose that a finite Frobenius group FH with cyclic kernel F and complement H acts by automorphisms on a Lie ring L in whose ground ring |F| is invertible. Let the fixed-point subring CL(H) of the complement be nilpotent of class c and the fixed-point subring of the kernel CL(F) be finite of order m.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

14 / 17

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SLIDE 58

Lie ring theorem

Lie ring Theorem (Khukhro, Makarenko, J. of Algebra 2013) Suppose that a finite Frobenius group FH with cyclic kernel F and complement H acts by automorphisms on a Lie ring L in whose ground ring |F| is invertible. Let the fixed-point subring CL(H) of the complement be nilpotent of class c and the fixed-point subring of the kernel CL(F) be finite of order m. Then L has a nilpotent Lie subring whose index in the additive group of L is bounded in terms of c, m, and |F| and whose nilpotency class is bounded in terms of c and |H| only.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

14 / 17

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SLIDE 59

Lie ring theorem, proof

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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SLIDE 60

Lie ring theorem, proof

The proof uses

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

15 / 17

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SLIDE 61

Lie ring theorem, proof

The proof uses the K–M–S theorem on Lie rings with fixed-point-free action of Frobenius kernel F, reformulated as a certain combinatorial fact about Lie rings with finite cyclic grading.

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

15 / 17

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SLIDE 62

Lie ring theorem, proof

The proof uses the K–M–S theorem on Lie rings with fixed-point-free action of Frobenius kernel F, reformulated as a certain combinatorial fact about Lie rings with finite cyclic grading. Method of graded centralizers. (This method was created by Khukhro for almost regular automorphisms of prime order and later developed in further studies of almost fixed-point-free automorphisms of Lie rings and nilpotent groups.)

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

15 / 17

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SLIDE 63

Frobenius groups of automorphisms: open problems

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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SLIDE 64

Frobenius groups of automorphisms: open problems

Mazurov’s question 17.72(b) remains open: Let GFH be a 2-Frobenius group (when GF is also a Frobenius group). Is the exponent of G bounded in terms of |H| and the exponent of CG(H)?

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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SLIDE 65

Frobenius groups of automorphisms: open problems

Mazurov’s question 17.72(b) remains open: Let GFH be a 2-Frobenius group (when GF is also a Frobenius group). Is the exponent of G bounded in terms of |H| and the exponent of CG(H)? Open problem. Let a finite Frobenius group FH with cyclic kernel F and complement H act by automorphisms on a finite group G. Suppose that the fixed-point subgroup CG(H) is soluble of derived length c and CG(F) = 1. Is the derived length of G bounded in terms

  • f that of CG(H) and |H|?
  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

16 / 17

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SLIDE 66

Frobenius groups of automorphisms: open problems

Mazurov’s question 17.72(b) remains open: Let GFH be a 2-Frobenius group (when GF is also a Frobenius group). Is the exponent of G bounded in terms of |H| and the exponent of CG(H)? Open problem. Let a finite Frobenius group FH with cyclic kernel F and complement H act by automorphisms on a finite group G. Suppose that the fixed-point subgroup CG(H) is soluble of derived length c and CG(F) = 1. Is the derived length of G bounded in terms

  • f that of CG(H) and |H|?

The last problem is already reduced to nilpotent groups, since it was proved by Khukhro that if CG(F) = 1, then the Fitting height of G is equal to that of CG(H).

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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SLIDE 67

Summary table

G FH is a Frobenius group; CG(F) = 1 FH is a Frobenius group; |CG(F)| = m |ϕ| = p is prime; CG(ϕ) = 1 |ϕ| is not prime; CG(ϕ) = 1 |G| Kh-M-Sh, 2010: |G| = |CG(H)||H|; Khukhro, 2013: restrictions on the order rk G rk G ≤ f(rk H, |H|); and the rank. nilpotency if H is nilpotent ⇒ G is nilpotent; M-Kh, 2013: F(G) has index bounded in terms in |F| and |CG(F)|; Tompson, 1959: G is nilpo- tent. There is a reduction to nilpotent groups. cl (G) If F is cyclic and CG(H) is nilpotent of class c ⇒ the nilpo- tency class of G is bounded in terms of c and |H|. If F is cyclic and CG(H) is nilpotent of class c ⇒ there is a char- acteristic subgroup G1 such that cl (G1) ≤ f(c, |H|), |G : G1| ≤ g(|FH|, |CG(F)|, c). Higman– Kreknin– Kostrikin, 1957, 1963: G is nilpo- tent of class bounded in terms of p. ????

  • N. Makarenko (Novosibirsk–Mulhouse)

Frobenius groups of automorphisms

  • St. Andrews, 2013

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