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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. Maria Tota Some finiteness conditions on centralizers or normalizers in groupsi , there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 jCG (x): hxij < 1 for all hxi 6 G. G is a BCI-group if, there exists a positive integer n such that jCG (x): hxij ≤ 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 = fa4 : a 2 Ag. 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 = fa4 : a 2 Ag. 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 = fa4 : a 2 Ag.