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Groups with Finiteness Conditions on Conjugates and Commutators

Groups with Finiteness Conditions on Conjugates and Commutators. Francesco de Giovanni Università di Napoli Federico II. A group G is called an FC-group if every element of G has only finitely many conjugates , or equivalently if the index |G :C G (x)| is finite for each element x

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Groups with Finiteness Conditions on Conjugates and Commutators

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  1. Groups with Finiteness Conditionson Conjugates and Commutators Francesco de Giovanni Università di Napoli Federico II

  2. A group G iscalledanFC-groupifeveryelementofG hasonlyfinitelymanyconjugates, or equivalentlyif the index|G:CG(x)|is finite foreachelementx Finite groups and abeliangroups are obviouslyexamplesofFC-groups Anydirectproductof finite or abeliansubgroupshas the propertyFC

  3. FC-groupshavebeenintroduced 70 years ago, and relevantcontributionshavebeengivenbyseveralimportantauthors R. Baer, P. Hall, B.H.Neumann, Y.M.Gorcakov, M.J.Tomkinson, L.A.Kurdachenko … and manyothers

  4. Clearlygroupswhosecentrehas finite index are FC-groups IfGis a group and xisanyelementofG, theconjugacyclassofx iscontained in the cosetxG’ ThereforeifG’ is finite, the group G hasboundedly finite conjugacyclasses

  5. Theorem 1 (B.H.Neumann, 1954) A group G hasboundedly finite conjugacyclassesif and onlyifitscommutatorsubgroup G’ is finite

  6. The relation between central-by-finite groups and finite-by-abelian groups is given by the following celebrated result Theorem 2 (Issai Schur, 1902) Let G be a group whose centre Z(G) has finite index. Then the commutator subgroup G’ of G is finite

  7. Theorem 3 (R. Baer, 1952) Let G be a group in which the term Zi(G) of the upper central series has finite index for some positive integer i. Then the (i+1)-th term γi+1(G) of the lower central series of G is finite

  8. Theorem 4 (P. Hall, 1956) Let G be a group such that the (i+1)-th term γi+1(G) of the lower central series of G is finite. Then the factor group G/Z2i(G) is finite

  9. Corollary A group G is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent

  10. The consideration of the locally dihedral 2-group shows that Baer’s teorem cannot be extended to terms with infinite ordinal type of the upper central series Similarly, free non-abelian groups show that Hall’s result does not hold for terms with infinite ordinal type of the lower central series

  11. Theorem 5 (M. De Falco – F. de Giovanni – C. Musella – Y.P. Sysak, 2009) A group G is finite over its hypercentre if and only if it contains a finite normal subgroup N such that G/N is hypercentral

  12. The propertiesC and C∞ A groupG has the propertyCif the set {X’ | X ≤ G} is finite A groupG has the propertyC∞if the set {X’ | X ≤ G, X infinite} is finite

  13. Tarskigroups (i.e. infinite simplegroupswhosepropernon-trivialsubgroupshave prime order) haveobviously the propertyC A groupG islocallygradedifeveryfinitelygeneratednon-trivialsubgroupofGcontains a propersubgroupof finite index Alllocally (soluble-by-finite) groups are locallygraded

  14. Theorem 6 (F. de Giovanni – D.J.S. Robinson, 2005) Let G be a locally graded group with the property C . Then the commutator subgroup G’ of G is finite

  15. The locally dihedral 2-group is a C∞-group with infinite commutator subgroup Let G be a Cernikov group, and let J be its finite residual (i.e. the largest divisible abelian subgroup of G). We say that G is irreducible if [J,G]≠{1} and J has no infinite proper K-invariant subgroups for CG(J)<K≤G

  16. Theorem 7 (F. de Giovanni – D.J.S. Robinson, 2005) Let G be a locallygradedgroupwith the propertyC∞. Theneither G’ is finite or G isanirreducibleCernikovgroup

  17. Recall that a group G is called metahamiltonian if every non-abelian subgroup of G is normal It was proved by G.M. Romalis and N.F. Sesekin that any locally graded metahamiltonian group has finite commutator subgroup

  18. In fact, Theorem 6 can beprovedalsoif the conditionCisimposedonlytonon-normalsubgroups Theorem 8 (F. De Mari – F. de Giovanni, 2006) Let G be a locallygradedgroupwithfinitelymanyderivedsubgroupsofnon-normalsubgroups. Then the commutatorsubgroup G’ of G is finite A similarremarkholdsalsofor the propertyC∞

  19. The properties K and K∞ A group G has the property K if for each element x of G the set {[x,H] | H ≤G} is finite A group G has the property K∞ if for each element x of G the set {[x,H] | H ≤ G, H infinite} is finite

  20. As the commutator subgroup of any FC-group is locally finite, it is easy to prove that all FC-groups have the property K On the other hand, also Tarski groups have the property K

  21. Theorem 9 (M. De Falco – F. de Giovanni – C. Musella, 2010) A group G isanFC-groupif and onlyifitislocally (soluble-by-finite) and has the propertyK

  22. Theorem 10 (M. De Falco – F. de Giovanni – C. Musella, 2010) A soluble-by-finite group G has the property K∞ if and only if it is either an FC-group or a finite extension of a group of type p∞ for some prime number p

  23. We shall say that a group G has the propertyN if for each subgroup X of G the set {[X,H] | H ≤G} is finite Theorem 11 (M. De Falco - F. de Giovanni – C. Musella, 2010) Let G be a soluble group with the property N . Then the commutator subgroup G’ of G is finite

  24. Let G be a group and let X be a subgroup of G. X is said to be inert in G if the index |X:X Xg| is finite for each element g of G X is said to be strongly inert in G if the index |X,Xg:X| is finite for each element g of G

  25. A group G is called inertial if all its subgroups are inert Similarly, G is strongly inertial if every subgroup of G is strongly inert

  26. The inequality |X:X Xg|≤ |X,Xg: Xg | proves that any strong inert subgroup of a group is likewise inert Thus strongly inertial groups are inertial It is easy to prove that any FC-group is strongly inertial

  27. Clearly, anynormalsubgroupofanarbitrarygroupis strong inert and so inert On the otherhand, finite subgroups are inertbutin generalthey are notstronglyinert In fact the infinite dihedralgroupisinertialbutitisnotstronglyinertial Note alsothatTarskigroups are inertial

  28. Theorem 12 (D.J.S. Robinson, 2006) Let G be a finitelygeneratedsoluble-by-finitegroup. Then G isinertialif and onlyifithasanabeliannormalsubgroup A of finite indexsuchthateveryelementof G induces on A apowerautomorphism In the samepaper Robinson alsoprovides a complete classificationofsoluble-by-finite minimax groupswhich are inertial

  29. A specialclassofstronglyinertialgroups: groups in whicheverysubgrouphas finite index in itsnormalclosure Theorem 13 (B.H.Neumann, 1955) In a group G everysubgrouphas finite index in itsnrmalclosureif and onlyif the commutatorsubgroup G’ of G is finite

  30. Neumann’s theorem cannot be extended to strongly inertial groups. In fact, the locally dihedral 2-group is strongly inertial but it has infinite commutator subgroup

  31. Theorem 14 (M. De Falco – F. de Giovanni – C. Musella – N. Trabelsi, 2010) Let G be a finitely generated strongly inertial group. Then the factor group G/Z(G) is finite

  32. As a consequence, the commutator subgroup of any strongly inertial group is locally finite Observe finally that strongly inertial groups can be completely described within the universe of soluble-by-finite minimax groups

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