Algebras, rings and groups with almostfixed-point-free automorphisms

Natalia Makarenko

Sobolev Institute of Mathematics, Novosibirsk, Universite de Haute Alsace

Conference in Honour of Evgeny Khukhro

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 1 / 1

Definitions

Let A be a group or an algebra over a field or a ring, ϕ ∈ Aut A.

Fixed-point-free au-tomorphism

=

automorphism without nontrivialfixed points, i.e.CA(ϕ) = {a ∈ A | aϕ = a} is triv-ial;

Almost fixed-point-free automorphism

= fixed-point subgroup CA(ϕ) isfinite;

= fixed-point subalgebra CA(ϕ) isfinite-dimensional;

= fixed-point subring CA(ϕ) is fi-nite;

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 2 / 1

Definitions: graded algebra

Let G be a group.

Def. A is a G-graded algebra over a field F if

A =⊕g∈G

Ag and AgAh ⊆ Agh,

where Ag are subspaces of a.

Def. Elements of Ag are referred to as homogeneous and thesubspaces Ag are called homogeneous components or gradingcomponents.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 3 / 1

Definitions: Lie type algebra

Def. An algebra A over a field F is called a Lie type algebra if for allelements a,b, c ∈ A there exist α, β ∈ F (depending on a,b, c) suchthat α 6= 0 and

[[a,b], c] = α[a, [b, c]] + β[[a, c],b].

Def. A G-graded algebra A = ⊕g∈GAg is called G-graded Lie typealgebra if it satisfies the following property: for all homogeneouselements a,b, c ∈ L there exist α, β ∈ F (depending on a,b, c) suchthat α 6= 0 and

[[a,b], c] = α[a, [b, c]] + β[[a, c],b]

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 4 / 1

Examples of algebras of Lie type

Examples

Associative algebras = algebras of Lie type with trivial grading;α = 1, β = 0.

(ab)c = αa(bc) + β(ac)b = a(bc).

Lie algebras = algebras of Lie type with trivial grading;α = 1, β = 1.

[[a,b], c] = α[a, [b, c]]+β[[a, c],b] = α[a, [b, c]]+β[[a, c],b] (Jacobi Identity);

[a,b] = −[b,a] (Anticommutativity).

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 5 / 1

Examples of algebras of Lie type

ExamplesLie superalgebras.

Def. A Lie superalgebra is (Z/2Z)-graded algebra L = L0⊕

L1with an operation [ . , . ], satisfying the following axioms:

[a,b] = −(−1)τθ[b,a];

[[a,b], c] = [a, [b, c]] + (−1)τθ[[a, c],b]for a ∈ Lτ , b ∈ Lθ.

Lie superalgebras are algebras of Lie type with α = 1,β(0,0) = β(0,1) = β(1,0) = 1, β(1,1) = −1.etc.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 6 / 1

Known results: fixed-point-free automorphisms offinite groups

Particular case: ϕ ∈ Aut G, |ϕ| is a prime, ϕ is fixe-point-freeautomorphism.

Higman, 1957. A nilpotent group G admitting afixed-point-free automorphism of prime order p isnilpotent of class bounded by a function h(p) de-pending only on p.

Higman, 1957 (Theorem on Lie rings). If a Lie ring L admits afixed-point-free automorphism of prime order p, then the nilpotencyclass of L is bounded by h(p).

Kreknin–Kostrikin, 1963: give another proof with a bound of h(p).N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 7 / 1

Known results: fixed-point-free automorphisms offinite groups

Thompson, 1959. A finite group admittinga fixed-point-free automorphism of primeorder is nilpotent.

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Fixed-point-free automorphisms of composite order:groups

Long standing conjecture . If a finite group G admits a fixed-point freeautomorphism of finite order n, then G is soluble of n-bounded derivedlength.

Remark.

There is a reduction to nilpotent groups (Clemens, 1976, reductionto soluble groups; Thompson, 1964, reduction to nilpotent groups).

The similar problem for Lie algebras is solved (Kreknin, 1963).

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 9 / 1

Fixed-point-free automorphisms of composite order:groups

Particular case: |ϕ| = 4.

Kovac, 1960 : If G is a locally finite (or locallynilpotent) group, and G admits a fixed-point-freeautomorphism ϕ ∈ Aut G, of order 4 then G(2) ≤Z (G) (in particular, G is solvable of length 3).

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 10 / 1

Known results: fixed-point-free automorphisms of Liealgebras

Borel & Mostov, 1955: If L is Liealgebra of finite dimension, ϕ ∈Aut L, ϕ is semisimple, ϕ is fixed-point-free

=⇒ L is soluble.

(The proof is based on the structural theory of Lie algebras.)

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 11 / 1

Known results: fixed-point-free automorphisms of Liealgebras

Kreknin, 1963: L is Lie algebra of arbitrary dimension, ϕ ∈ Aut ,Lϕn = 1, CL(ϕ) = 0.

=⇒ L is soluble of derived length ≤ 2n − 2.

Remark. The condition ϕn = 1 is essential.

A virtually equivalent statement of Kreknin’s theorem in terms ofgraded Lie algebras:

Kreknin, 1963: Let L =∑n−1

i=0 Li be a (Z/nZ)-graded algebra.

If L0 = 0 =⇒ L is soluble of derived length ≤ 2n − 2.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 12 / 1

Fixed-point-free automorphisms: associative algebras

Bergman & Isaacs, 1972: Let Abe an associative algebra over afield F , G ≤ Aut A, G finite,char F - |G|.

If CA(G) = 0 =⇒ A is nilpo-tent.

Remarks:The nilpotency class of A is boundedby a function depending only on |G|.

More generally, if CA(G) is nilpotentthen A is also nilpotent.

The restrictions on G are essential.

Example.N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 13 / 1

Almost fixed-point-free automorphisms: groups,automorphism of prime order

Particular case: |ϕ| = 2.

Brauer, Fowler, 1955. If a finitegroup admits an automorphism oforder 2 and the number of fixedpoints is m, then G has a solvablesubgroup of index bounded by afunction depending only on m.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 14 / 1

Almost fixed-point-free automorphisms: groups,automorphism of prime order

Fong (1976): reduction to soluble groups (moduloclassification).

+ Hartley–Meixner (1981): reductionto nilpotent groups.

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Almost fixed-point-free automorphisms: groups,automorphism of prime order

Khukhro, 1992. Let G be a fi-nite group, p a prime number; ϕ ∈Aut G, ϕp = 1. |CG(ϕ)| = m.=⇒ G contains a nilpotent nor-mal subgroup of p-bounded nilpo-tency class and of (p,m)-boundedindex.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 16 / 1

Almost fixed-point-free automorphisms: groups,automorphism of arbitrary order

Particular case: |ϕ| = 4.

Khukhro–Makarenko, 2005. LetG be a finite group, ϕ ∈ Aut G,ϕ4 = 1. |CG(ϕ)| = m.

=⇒ G contains a char-acteristic soluble subgroup ofm-bounded index and of d.l.bounded by a constant.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 17 / 1

Almost fixed-point-free automorphisms: groups,arbitrary order

Particular case: |ϕ| = 4.

Khukhro–Makarenko, 2005. LetG be a finite group, ϕ ∈ Aut G,ϕ4 = 1. |CG(ϕ)| = m.

=⇒ G contains a char-acteristic soluble subgroup ofm-bounded index and of d.l.bounded by a constant.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 18 / 1

Almost fixed-point-free automorphisms: Lie algebras

Particular case: |ϕ| = p is prime.

Khukhro, 1992. Let L be a Lie ring, p a prime number; ϕ ∈ Aut L,ϕp = 1. |CL(ϕ)| = m, pL = L =⇒ L has a nilpotent normal subringof p-bounded nilpotency class and of (p,m)-bounded index.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 19 / 1

Almost fixed-point-free automorphisms: Lie algebras

General case.

Makarenko & Khukhro, 2004 Let Lbe a Lie algebra (of arbitrary dimen-sion); ϕ ∈ Aut L, ϕn = 1.

If CL(ϕ) is finite-dimensional =⇒L is almost soluble, i.e. L hasa soluble ideal H of finite codimen-sion.

Moreover, the derived length of H is bounded by a functiondepending only on n, and the codimension of H is bounded by afunction on dimCL(ϕ) and n.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 20 / 1

Almost fixed-point-free automorphisms: graded Liealgebras

Makarenko & Khukhro, 2004 Let L =∑n−1

i=0 Li be a Z/nZ-graded Liealgebra.

If L0 is finite-dimensional =⇒ the same as in the previoustheorem:

L has soluble ideal H of finite codimension;Moreover, the derived length of H is bounded by a functiondepending only on n and the codimension of H is bounded by afunction on n and dim L0.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 21 / 1

Other results

+ +

Bahturin–Zaicev–Linchenko, 1997, 1998: Let L =⊕

g∈G Lg be a Lie

algebra over an arbitrary field F , G finite group. If the unityhomogeneous component L1 satisfies a non-trivial identity, then Lalso satisfies a non-trivial identity.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 22 / 1

Main result: graded algebras of Lie type

Theorem 1. (Makarenko, 2015, J. of Algebra). Let

L =n−1⊕i=0

Li

be a Z/nZ-graded Lie type algebra over a field.

If L0 has finite dimension, then L is almost soluble, i.e. L has asoluble ideal M of finite codimension. Moreover, the derived length ofM is bounded by a function of n, and the codimension of M isbounded by a function of n and dim L0.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 23 / 1

Connection with almost fixed-point-freeautomorphisms

Let A be an arbitrary algebra over a field F , ϕ ∈ Aut A, ϕn = 1,char F - n. Suppose that CA(ϕ) = {x ∈ A | xϕ = x} = 0 ordim CA(ϕ) = m.

The ground field can be extended by a primitive n th root of unity,ω = n

√1.

The eigenspacesAi = {x | xϕ = ωix}

satisfy the inclusions:AiAj ⊆ Ai+j(mod n).

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Connection with almost fixed-point-freeautomorphisms

In addition,A = nA = A0 ⊕ A1 ⊕ A2 . . .⊕ An−1.

By definition, A0 = CA(ϕ).

If CA(ϕ) = 0, then A0 = 0.

If dim CA(ϕ) = m is finite, then dim A0 is also finite.

The problem reduces to Z/nZ-graded algebra∑n−1

i=0 Ai with trivial orfinite-dimensional component A0.

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Main results: algebra of Lie type with an almostfixed-point-free automorphism

Theorem 2. (Makarenko, 2015, J. of Algebra). Let L be a Lie type (ofpossibly infinite dimension) over an arbitrary field, ϕ ∈ Aut L, ϕn = 1.CL(ϕ) has finite dimension m. =⇒ L is almost soluble, i.e. L has

a soluble ideal M of finite codimension.

Moreover, the derived length of M is bounded by a function of n, andthe codimension of M is bounded by a function of n and dim L0.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 26 / 1

Some ideas of the proof of Theorem 1

The proof is based on the same ideas that we used in analogous resulton Lie algebras.

The proof of this previous result is of purely combinatorial nature anduses only the Jacobi identity and the identity of anticommutativity.

In our case the Jacobi identity is successfully substituted by the gradedidentity

a(bc) = α(ab)c + β(ac)b.

The main difficulty is the absence of the identity of anticommutativity.

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 27 / 1

Applications

to associative algebras with an almost fixed-point-freeautomorphism (generalization of the Bergman– Isaacs theoremon fixed-point-free authomorphism)... (new result);

to graded Lie algebras...(known result, theoremMakarenko–Khukhro, 2004).

to Lie superalgebras with an almost fixed-point-free automorphism(new result).

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 28 / 1

Application of Theorem 2 to Lie superalgebras

Def. An automorphism ϕ : L→ L of Lie superalgebra preserves thegrading: Lϕα ⊆ Lα.

Corollary.

Let L = L0 ⊕ L1 be a Lie superalgebra, ϕ an automorphism of finiteodd order n. Suppose that char F does not divide n.

If dimCL0(ϕ) = m <∞, =⇒ L has a homogeneous solubleideal H of derived length ≤ g(n) and codimension ≤ f (m,n).

If dimCL0(ϕ) = 0 (ϕ is fixed-point-free on L0), =⇒ L is solubleof derived length ≤ g(n).

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 29 / 1

Main results: almost fixed-point-free automorphisms ofassociative algebras

Corollary of theorem 2

Let A be an associative algebra over a field F ; ϕ ∈ Aut (A), ϕn = 1.

If CA(ϕ) is finite-dimensional and char F - n =⇒ A is almostnilpotent, i.e. A has nilpotent ideal H of finite codimension...

Theorem 3 (Makarenko, 2016). Let A be an associative algebra over afield F , G ≤ Aut A, G a finite soluble group of order n. Suppose thatchar F - n.

If CA(G) is finite-dimensional of dimension m, then A has a nilpotentideal of nilpotency index bounded by a function of n and of finitecodimension bounded by a function of m and n.

Open problem. For an arbitrary finite group of automorphisms?

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Some ideas of the proof for associative algebras

Theorem 4 (Makarenko, 2016). Let G be a finite group of order n andlet

A =⊕g∈G

Ag

be a G-graded associative algebra over a field F .

If the identity component Ae has an ideal Ie (in Ae) of finitecodimension m in Ae and Id

e = 0, then A is almost nilpotent. Moreover,

A has a homogeneous ideal H such that H f (n,d) = 0 for some functionf and codimension of H is bounded by a function on n, d and m.

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Lemma (Bergman–Isaacs Theorem). Let G be afinite group of automorphisms of an associativealgebra A of order n. If A has no n-torsion andCA(G) is nilpotent of index d , then A is nilpotent ofindex at most hd , where h = 1 +

∏ni=0(C

in + 1).

Lemma (Kharchenko) Let G be a finite group of automor-phisms of an associative algebra A of order n. If A has non-torsion and the fixed-point subalgebra CA(G) containsa nilpotent ideal I C CA(G) of nilpotency index d , thenA contains a G-invariant ideal J ≥ I of nilpotency indexbounded by a function of n and d .

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 32 / 1

A

|ϕ| = p

ϕ is f-p-f.

|ϕ| <∞

ϕ is f-p-f.

|ϕ| = pϕ is almostf-p-f

|ϕ| <∞ϕ is almost f-p-f.

Finitegroups

Thompson, 59:A is nilpotent;Higman-Kreknin-Kostrikin (57, 63):cl(A) ≤ h(p)

reduction to nilpo-tent group;Kovacs, 60: |ϕ| =4.Open question

Khukhro, 92:A has a normalnilpotent sub-group B withbounds for cl(B)and index of B.

Makarenko-Khukhro(2005): |ϕ| = 4Open question.

Liealgebras

Higman-Kreknin-Kostrikin (57, 63):cl(A) ≤ h(p).

Kreknin,1963:A is soluble ofd.l.≤ f (n).

Khukhro, 92:A has a nilpotentsubring B withbounds for cl(B)and codim. of B.

Makarenko–Khukhro, 2004:A is almostsoluble.

Assoc.algebras

Bergman–Isaacs,72:A is nilpotent.

Bergman–Isaacs,72:A is nilpotent.

??? ???

Liesuper-algebras

??? ??? ??? ???

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A

|ϕ| = p

ϕ is f-p-f.

|ϕ| <∞

ϕ is f-p-f.

|ϕ| = pϕ is almostf-p-f

|ϕ| <∞ϕ is almost f-p-f.

Finitegroups

Thompson, 59:A is nilpotent;Higman-Kreknin-Kostrikin (57, 63):cl(A) ≤ h(p)

reduction to nilpo-tent group;Kovacs, 60: |ϕ| =4.Open question

Khukhro, 92:A has a normalnilpotent sub-group B withbounds for cl(B)and index of B.

Makarenko-Khukhro(2005): |ϕ| = 4Open question.

Liealgebras

Higman-Kreknin-Kostrikin (57, 63):cl(A) ≤ h(p).

Kreknin,1963:A is soluble ofd.l.≤ f (n).

Khukhro, 92:A has a nilpotentsubring B withbounds for cl(B)and codim. of B.

Makarenko–Khukhro, 2004:A is almostsoluble.

Assoc.algebras

Bergman–Isaacs,72:A is nilpotent.

Bergman–Isaacs,72:A is nilpotent.

Makarenko, 16: Ais almost nilpo-tent.

Makarenko,16: A is almostnilpotent.

Liesuper-algebras

A is soluble A is soluble A is almostsoluble

A is almostsoluble

N. Makarenko (Novosibirsk–Mulhouse) Almost fixed-point-free automorphisms Lincoln, 2016 34 / 1