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`adegli Studi di Salerno Dipartimento di Matematica

“GtG Summer 2017” Trento, 17 giugno 2017

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

|CG (x): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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

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 |CG (x): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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

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 G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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

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): hxi| < ∞ for all hxi 6 G.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi , there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 G.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 G.

Examples G is a BCI-group =⇒ G is an FCI-group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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 A is abelian of finite 2-rank =⇒ Dih(A) is a BCI-group G is a Tarski monster group =⇒ G is a BCI-group

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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 G is a Tarski monster group =⇒ G is a BCI-group

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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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 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): hxi| < ∞ for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that

|CG (x): hxi| ≤ n for all hxi 6 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 ... 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!

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 A non abelian free group is an FCI-group which is not a BCI-group!

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 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 G is an FCI-(BCI-)group, H ≤ G =⇒ H is an FCI-(BCI-)group G is an FCI-(BCI-)group, N / G 6=⇒ 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

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 G is an FCI-(BCI-)group, N / G 6=⇒ 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

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 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

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 6=⇒ G/N is an FCI-(BCI-)group

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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

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 6=⇒ 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 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

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 6=⇒ 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 G periodic, FCI-(BCI-)group =⇒ G/Z(G) is an FCI-(BCI-)group

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 6=⇒ 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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 6=⇒ 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 G is an FCI-group ⇐⇒ |CG (x)| < ∞ for all hxi 6 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.

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

Let G be periodic.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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.

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 hxi 6 G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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.

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 hxi 6 G

Theorem [Shunkov]

Let G be periodic, x ∈ G, |x| = 2, |CG (x)| < ∞.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi Theorem [Meixner, Hartley, Pattet, Khukhro]

Let G be locally finite, x ∈ G, |x| = p, |CG (x)| < ∞. Then, G is nilpotent-by-finite.

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 hxi 6 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 Then, G is nilpotent-by-finite.

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 hxi 6 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 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 hxi 6 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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 : hxi / 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): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 Theorem Let G be a locally finite group satisfying (*). Then |G : FC(G)| is finite.

G periodic, FCI-group =⇒ FC(G) = {x ∈ G : hxi / 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): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 Then |G : FC(G)| is finite.

G periodic, FCI-group =⇒ FC(G) = {x ∈ G : hxi / 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): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 G periodic, FCI-group =⇒ FC(G) = {x ∈ G : hxi / 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): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 G periodic, FCI-group =⇒ G satisfies (*)

De Falco, de Giovanni, Musella, Trabelsi (2017)

G is an AFC-group ⇔ x ∈ FC(G) or |CG (x): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 : hxi / G}

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi De Falco, de Giovanni, Musella, Trabelsi (2017)

G is an AFC-group ⇔ x ∈ FC(G) or |CG (x): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 : hxi / G} G periodic, FCI-group =⇒ G satisfies (*)

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi G locally finite, FCI-group =⇒ |G : FC(G)| finite

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 : hxi / 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): hxi| finite, ∀x ∈ G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 : hxi / 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): hxi| finite, ∀x ∈ G

G locally finite, FCI-group =⇒ |G : FC(G)| finite

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 ∼ or Q = Q8. G is an infinite, locally finite, FCI-group iff G = D, or G = hD, xi, D of finite 2-rank, x acts on D as a power automorphism and there exists m > 1 such that xm ∈ D and k |CA(x )| is finite, ∀k = 1,..., m − 1.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 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.

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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi Tarski monster groups are periodic BCI-groups, which are not locally graded.

Theorem Every locally graded periodic BCI-group is locally finite.

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 Theorem Every locally graded periodic BCI-group is locally finite.

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 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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi Examples There exist finitely generated infinite periodic groups which are residually finite but not FCI (Golod, Grigorchuk and Gupta-Sidki).

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

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 of 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).

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi |NG (H): H| < ∞ for all H 6 G. (1)

G is a BNI-group if, there exists a positive integer n such that

|NG (H): H| ≤ n for all H 6 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

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

G is an FNI-group if

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi G is a BNI-group if, there exists a positive integer n such that

|NG (H): H| ≤ n for all H 6 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

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

G is an FNI-group if

|NG (H): H| < ∞ for all H 6 G. (1)

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi there exists a positive integer n such that

|NG (H): H| ≤ n for all H 6 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

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

G is an FNI-group if

|NG (H): H| < ∞ for all H 6 G. (1)

G is a BNI-group if,

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

G is an FNI-group if

|NG (H): H| < ∞ for all H 6 G. (1)

G is a BNI-group if, there exists a positive integer n such that

|NG (H): H| ≤ n for all H 6 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

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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

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 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 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 = hg, Ai is a 2-group, where A is infinite abelian of finite rank, and g is an element of order at most 4 such that g 2 ∈ A and ag = a−1 for all a ∈ A. 2 Q is a finite abelian 20-group.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 6= 1.

3 If p = 2 and D2 is infinite, then t2 6= 1, −1. Then the semidirect product G = hgi n D, where g is of infinite order 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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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!

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 hxi 6 G. 3 Either G is abelian or G = hg, Ai, where A is a non-periodic abelian group of finite 2-rank and g is an element of order at most 4 such that g 2 ∈ A and ag = a−1 for all a ∈ A.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi There exist non-periodic locally nilpotent FCI-groups, which are not BCI-groups!

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 hxi 6 G. 3 Either G is abelian or G = hg, Ai, where A is a non-periodic abelian group of finite 2-rank and g is an element of order at most 4 such that g 2 ∈ 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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 hxi 6 G. 3 Either G is abelian or G = hg, Ai, where A is a non-periodic abelian group of finite 2-rank and g is an element of order at most 4 such that g 2 ∈ 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!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 = hg, Ai, 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 g 2 ∈ A and ag = a−1 for all a ∈ A.

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 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 = hg, Ai, 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 g 2 ∈ A and ag = a−1 for all a ∈ A.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi Conditions on centralizers Locally finite case Conditions on normalizers Locally nilpotent case

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

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi (∀ x ∈/ D).

Sketch of the proof: Let G be f. g. and assume, by contradiction, G infinite. Let n ≥ 1 such that |CG (x): hxi| ≤ n for all hxi 6 G Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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

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

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi (∀ x ∈/ D).

Let n ≥ 1 such that |CG (x): hxi| ≤ n for all hxi 6 G Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi (∀ x ∈/ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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): hxi| ≤ n for all hxi 6 G

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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): hxi| ≤ n for all hxi 6 G (∀ x ∈/ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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): hxi| ≤ n for all hxi 6 G (∀ x ∈/ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order pq.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi G/R f. g., res. fin., exp(G/R) < ∞ ⇒ G/R finite ⇒ 1 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

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): hxi| ≤ n for all hxi 6 G (∀ x ∈/ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order pq. R the finite residual of G ⇒ π(G/R) finite ⇒ exp(G/R) < ∞.

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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): hxi| ≤ n for all hxi 6 G (∀ x ∈/ D). Set D := FC(G) ⇒ D finite ⇒ G/D infinite locally graded BCI. It follows |CG (x): hxi| ≤ n for all x ∈ G r {1}. So, each finite quotient of G is soluble and has not elements of order 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 6= R f. g. Then, ∃ K < R : |R : K| < ∞ ⇒ |G : K| < ∞ ⇒ R ≤ K. Contradiction!

Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi 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