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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case Some finiteness conditions on centralizers or normalizers in groups Maria Tota (joint work with G.A. Fern andez-Alcober, L. Legarreta and A.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Some finiteness conditions on centralizers or normalizers in groups

Maria Tota (joint work with G.A. Fern´ andez-Alcober,

L. Legarreta and A. Tortora)

Universit` a degli Studi di Salerno Dipartimento di Matematica

“GtG Summer 2017” Trento, 17 giugno 2017

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI G Dedekind = ⇒ G is BCI

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI G Dedekind = ⇒ G is BCI A is abelian of finite 2-rank = ⇒ Dih(A) is a BCI-group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI G Dedekind = ⇒ G is BCI A is abelian of finite 2-rank = ⇒ Dih(A) is a BCI-group G is a Tarski monster group = ⇒ G is a BCI-group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Also F is a free group = ⇒ F is an FCI-group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Also F is a free group = ⇒ F is an FCI-group ... If G is a torsion free BCI-group, then G is abelian.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Also F is a free group = ⇒ F is an FCI-group ... If G is a torsion free BCI-group, then G is abelian. A non abelian free group is an FCI-group which is not a BCI-group!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}. Then G =Dih(A) is a BCI-group but G/N IS NOT!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}. Then G =Dih(A) is a BCI-group but G/N IS NOT! Proposition Let G be an FCI-(BCI-)group and N ⊳ G, N finite. Then G/N is an FCI-(BCI-)group.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}. Then G =Dih(A) is a BCI-group but G/N IS NOT! Proposition Let G be an FCI-(BCI-)group and N ⊳ G, N finite. Then G/N is an FCI-(BCI-)group. G periodic, FCI-(BCI-)group = ⇒ G/Z(G) is an FCI-(BCI-)group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞. Then, G is locally finite (and soluble-by-finite).

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞. Then, G is locally finite (and soluble-by-finite). Theorem [Meixner, Hartley, Pattet, Khukhro] Let G be locally finite, x ∈ G, |x| = p, |CG(x)| < ∞.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞. Then, G is locally finite (and soluble-by-finite). Theorem [Meixner, Hartley, Pattet, Khukhro] Let G be locally finite, x ∈ G, |x| = p, |CG(x)| < ∞. Then, G is nilpotent-by-finite.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying ().

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G}

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G} G periodic, FCI-group = ⇒ G satisfies (*)

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G} G periodic, FCI-group = ⇒ G satisfies (*) De Falco, de Giovanni, Musella, Trabelsi (2017) G is an AFC-group ⇔ x ∈ FC(G) or |CG(x) : x| finite, ∀x ∈ G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G} G periodic, FCI-group = ⇒ G satisfies (*) De Falco, de Giovanni, Musella, Trabelsi (2017) G is an AFC-group ⇔ x ∈ FC(G) or |CG(x) : x| finite, ∀x ∈ G G locally finite, FCI-group = ⇒ |G : FC(G)| finite

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem [Fern´ andez-Alcober, Legarreta, Tortora, T.] Let D = Q × A an infinite, periodic, Dedekind group, where Q ∼ = 1

r Q ∼

= Q8. G is an infinite, locally finite, FCI-group iff G = D, or G = D, x, D of finite 2-rank, x acts on D as a power automorphism and there exists m > 1 such that xm ∈ D and |CA(xk)| is finite, ∀k = 1, . . . , m − 1.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Corollary Let G be a locally finite group. Then the following facts are equivalent:

1 G is an FCI-group 2 G is a BCI-group Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. Given a prime p. Tarski monster groups are infinite (simple) p-groups, all of whose proper non.trivial subgroups are of order p.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. Given a prime p. Tarski monster groups are infinite (simple) p-groups, all of whose proper non.trivial subgroups are of order p. Tarski monster groups are periodic BCI-groups, which are not locally graded.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. Given a prime p. Tarski monster groups are infinite (simple) p-groups, all of whose proper non.trivial subgroups are of order p. Tarski monster groups are periodic BCI-groups, which are not locally graded. Theorem Every locally graded periodic BCI-group is locally finite.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Questions Does the previous theorem hold for FCI-groups? Given a periodic residually finite group G in which the centralizer

f each non-trivial element is finite, is G finite?

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Questions Does the previous theorem hold for FCI-groups? Given a periodic residually finite group G in which the centralizer

f each non-trivial element is finite, is G finite?

Examples There exist finitely generated infinite periodic groups which are residually finite but not FCI (Golod, Grigorchuk and Gupta-Sidki).

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

G is an FNI-group if

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1)

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G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1) G is a BNI-group if,

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1) G is a BNI-group if, there exists a positive integer n such that |NG(H) : H| ≤ n for all H ⋪ G. (2) G is a BNI-group ⇒ G is an FNI-group G is an FNI-group ⇒ G is an FCI-group G is a BNI-group ⇒ G is a BCI-group

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be locally finite. Then the following facts are equivalent:

1 G is an FNI-group 2 G is an FCI-group 3 G is a BNI-group 4 G is a BCI-group Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be locally finite. Then the following facts are equivalent:

1 G is an FNI-group 2 G is an FCI-group 3 G is a BNI-group 4 G is a BCI-group

Corollary Let G be locally graded and periodic. Then the following facts are equivalent:

1 G is a BNI-group 2 G is a BCI-group Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Classification Theorem (periodic case) Let G be a non-Dedekind infinite periodic group. Then G is a locally nilpotent FCI-group if and only if G = P × Q, where P and Q are as follows:

1 P = g, A is a 2-group, where A is infinite abelian of finite

rank, and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

2 Q is a finite abelian 2′-group. Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Classification Theorem (non periodic case) Let D be a periodic Dedekind group such that π(D) is finite and Dp is of finite rank for every p ∈ π(D). Let ϕ be a power automorphism of D, and write tp = exp ϕp whenever Dp is abelian. Assume that the following conditions hold:

1 If D2 is non-abelian, ϕ2 is the identity automorphism. 2 If p > 2 then tp ≡ 1 (mod p), and if Dp is infinite also tp = 1. 3 If p = 2 and D2 is infinite, then t2 = 1, −1.

Then the semidirect product G = g ⋉ D, where g is of infinite

rder and acts on D via ϕ, is a locally nilpotent FCI-group.

Conversely, every non-periodic locally nilpotent FCI-group is either abelian or isomorphic to a group as above, D being the torsion subgroup of G.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

1 G is a BCI-group. 2 There exists n ∈ N such that |CG(x)| ≤ n whenever x ⋪ G. 3 Either G is abelian or G = g, A, where A is a non-periodic

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

1 G is a BCI-group. 2 There exists n ∈ N such that |CG(x)| ≤ n whenever x ⋪ G. 3 Either G is abelian or G = g, A, where A is a non-periodic

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A. Corollary Let G be a non-periodic locally nilpotent group. If G is a BCI-group then G is abelian.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

1 G is a BCI-group. 2 There exists n ∈ N such that |CG(x)| ≤ n whenever x ⋪ G. 3 Either G is abelian or G = g, A, where A is a non-periodic

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A. Corollary Let G be a non-periodic locally nilpotent group. If G is a BCI-group then G is abelian. There exist non-periodic locally nilpotent FCI-groups, which are not BCI-groups!

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be a non-periodic locally nilpotent group. Then the following conditions are equivalent:

1 G is an FNI-group 2 G is an FCI-group Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem Let G be a non-periodic locally nilpotent group. Then the following conditions are equivalent:

1 G is an FNI-group 2 G is an FCI-group

Theorem Let G be a non-periodic group. Then the following hold: G is a BNI-group if and only if either G is abelian or G = g, A, where A is a non-periodic abelian group of finite 0-rank and finite 2-rank, and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

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[Robinson (2016)] FCI-groups and FNI-groups have been classified in the locally (soluble-by-finite) case.

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite.

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G (∀ x / ∈ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI.

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G (∀ x / ∈ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG(x) : x| ≤ n for all x ∈ G {1}. So, each finite quotient of G is soluble and has not elements of

rder pq.

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Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G (∀ x / ∈ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG(x) : x| ≤ n for all x ∈ G {1}. So, each finite quotient of G is soluble and has not elements of

rder pq.

R the finite residual of G ⇒ π(G/R) finite ⇒ exp(G/R) < ∞.

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Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G (∀ x / ∈ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG(x) : x| ≤ n for all x ∈ G {1}. So, each finite quotient of G is soluble and has not elements of

rder pq.

R the finite residual of G ⇒ π(G/R) finite ⇒ exp(G/R) < ∞. G/R f. g., res. fin., exp(G/R) < ∞ ⇒ G/R finite ⇒ 1 = R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

SLIDE 65

Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

Thank you!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi

Some finiteness conditions on centralizers or normalizers in groups

Maria Tota (joint work with G.A. Fern´ andez-Alcober,

Universit` a degli Studi di Salerno Dipartimento di Matematica

“GtG Summer 2017” Trento, 17 giugno 2017

Let G be a group.

Let G be a group. G is an FCI-group if

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G.

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G.

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI

Let G be a group. G is an FCI-group if |CG(x) : x| < ∞ for all x ⋪ G. G is a BCI-group if, there exists a positive integer n such that |CG(x) : x| ≤ n for all x ⋪ G. Examples G is a BCI-group = ⇒ G is an FCI-group G finite = ⇒ G is BCI G Dedekind = ⇒ G is BCI

Also F is a free group = ⇒ F is an FCI-group

Also F is a free group = ⇒ F is an FCI-group ... If G is a torsion free BCI-group, then G is abelian.

Also F is a free group = ⇒ F is an FCI-group ... If G is a torsion free BCI-group, then G is abelian. A non abelian free group is an FCI-group which is not a BCI-group!

Closure properties

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}.

Closure properties G is an FCI-(BCI-)group, H ≤ G = ⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N ⊳ G = ⇒ G/N is an FCI-(BCI-)group Counterexample Let A torsion free, abelian of infinite 0-rank and N = {a4 : a ∈ A}. Then G =Dih(A) is a BCI-group but G/N IS NOT!

Let G be periodic.

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞.

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞. Then, G is locally finite (and soluble-by-finite).

Let G be periodic. G is an FCI-group ⇐ ⇒ |CG(x)| < ∞ for all x ⋪ G Theorem [Shunkov] Let G be periodic, x ∈ G, |x| = 2, |CG(x)| < ∞. Then, G is locally finite (and soluble-by-finite). Theorem [Meixner, Hartley, Pattet, Khukhro] Let G be locally finite, x ∈ G, |x| = p, |CG(x)| < ∞.

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G.

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (*).

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (*). Then |G : FC(G)| is finite.

Shalev (1994) G satisfies (*) iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (*). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G}

Theorem [Fern´ andez-Alcober, Legarreta, Tortora, T.] Let D = Q × A an infinite, periodic, Dedekind group, where Q ∼ = 1

= Q8. G is an infinite, locally finite, FCI-group iff G = D, or G = D, x, D of finite 2-rank, x acts on D as a power automorphism and there exists m > 1 such that xm ∈ D and |CA(xk)| is finite, ∀k = 1, . . . , m − 1.

Corollary Let G be a locally finite group. Then the following facts are equivalent:

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image.

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. Given a prime p. Tarski monster groups are infinite (simple) p-groups, all of whose proper non.trivial subgroups are of order p.

Questions Does the previous theorem hold for FCI-groups? Given a periodic residually finite group G in which the centralizer

Questions Does the previous theorem hold for FCI-groups? Given a periodic residually finite group G in which the centralizer

Examples There exist finitely generated infinite periodic groups which are residually finite but not FCI (Golod, Grigorchuk and Gupta-Sidki).

G is an FNI-group if

G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1)

G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1) G is a BNI-group if,

G is an FNI-group if |NG(H) : H| < ∞ for all H ⋪ G. (1) G is a BNI-group if, there exists a positive integer n such that |NG(H) : H| ≤ n for all H ⋪ G. (2) G is a BNI-group ⇒ G is an FNI-group G is an FNI-group ⇒ G is an FCI-group G is a BNI-group ⇒ G is a BCI-group

Theorem Let G be locally finite. Then the following facts are equivalent:

Theorem Let G be locally finite. Then the following facts are equivalent:

Corollary Let G be locally graded and periodic. Then the following facts are equivalent:

Classification Theorem (periodic case) Let G be a non-Dedekind infinite periodic group. Then G is a locally nilpotent FCI-group if and only if G = P × Q, where P and Q are as follows:

rank, and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

Classification Theorem (non periodic case) Let D be a periodic Dedekind group such that π(D) is finite and Dp is of finite rank for every p ∈ π(D). Let ϕ be a power automorphism of D, and write tp = exp ϕp whenever Dp is abelian. Assume that the following conditions hold:

Then the semidirect product G = g ⋉ D, where g is of infinite

Conversely, every non-periodic locally nilpotent FCI-group is either abelian or isomorphic to a group as above, D being the torsion subgroup of G.

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A. Corollary Let G be a non-periodic locally nilpotent group. If G is a BCI-group then G is abelian.

Theorem Let G be a non-periodic group. Then the following conditions are equivalent:

abelian group of finite 2-rank and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A. Corollary Let G be a non-periodic locally nilpotent group. If G is a BCI-group then G is abelian. There exist non-periodic locally nilpotent FCI-groups, which are not BCI-groups!

Theorem Let G be a non-periodic locally nilpotent group. Then the following conditions are equivalent:

Theorem Let G be a non-periodic locally nilpotent group. Then the following conditions are equivalent:

Theorem Let G be a non-periodic group. Then the following hold: G is a BNI-group if and only if either G is abelian or G = g, A, where A is a non-periodic abelian group of finite 0-rank and finite 2-rank, and g is an element of order at most 4 such that g2 ∈ A and ag = a−1 for all a ∈ A.

[Robinson (2016)] FCI-groups and FNI-groups have been classified in the locally (soluble-by-finite) case.

Theorem G locally graded periodic BCI-group = ⇒ G locally finite

Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite.

Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G

Theorem G locally graded periodic BCI-group = ⇒ G locally finite Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG(x) : x| ≤ n for all x ⋪ G (∀ x / ∈ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI.

R the finite residual of G ⇒ π(G/R) finite ⇒ exp(G/R) < ∞.

R the finite residual of G ⇒ π(G/R) finite ⇒ exp(G/R) < ∞. G/R f. g., res. fin., exp(G/R) < ∞ ⇒ G/R finite ⇒ 1 = R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

Thank you!

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying ().

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite.

Shalev (1994) G satisfies () iff |CG(x)| finite or |G : CG(x)| finite, for all x ∈ G. FC(G) = {x ∈ G : [x]Cg is finite} FC-centre of G Theorem Let G be a locally finite group satisfying (). Then |G : FC(G)| is finite. G periodic, FCI-group = ⇒ FC(G) = {x ∈ G : x ⊳ G}