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Groups with Finiteness Conditions on Conjugates and Commutators Francesco de Giovanni Università di Napoli Federico II

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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 Finite groups and abelian groups are obviously examples of FC -groups Any direct product of finite or abelian subgroups has the property FC

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FC -groups have been introduced 70 years ago, and relevant contributions have been given by several important authors R. Baer, P. Hall, B.H. Neumann, Y.M. Gorcakov, M.J. Tomkinson, L.A. Kurdachenko … and many others

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Clearly groups whose centre has finite index are FC -groups If G is a group and x is any element of G, the conjugacy class of x is contained in the coset xG’ Therefore if G’ is finite, the group G has boundedly finite conjugacy classes

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Theorem 1 (B.H. Neumann, 1954) A group G has boundedly finite conjugacy classes if and only if its commutator subgroup G’ is finite

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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

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

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Theorem 4 (P. Hall, 1956) Let G be a group such that the (i+1)-th term γ i+1 (G) of the lower central series of G is finite. Then the factor group G/Z 2i (G) is finite

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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

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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

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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

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The properties C and C ∞ A group G has the property C if the set { X’ | X ≤ G} is finite A group G has the property C ∞ if the set { X’ | X ≤ G, X infinite} is finite

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Tarski groups (i.e. infinite simple groups whose proper non-trivial subgroups have prime order) have obviously the property C A group G is locally graded if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index All locally (soluble-by-finite) groups are locally graded

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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

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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 C G (J)

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Theorem 7 (F. de Giovanni – D.J.S. Robinson, 2005) Let G be a locally graded group with the property C ∞. Then either G’ is finite or G is an irreducible Cernikov group

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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

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In fact, Theorem 6 can be proved also if the condition C is imposed only to non-normal subgroups Theorem 8 (F. De Mari – F. de Giovanni, 2006) Let G be a locally graded group with finitely many derived subgroups of non-normal subgroups. Then the commutator subgroup G’ of G is finite A similar remark holds also for the property C ∞

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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

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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

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Theorem 9 (M. De Falco – F. de Giovanni – C. Musella, 2010) A group G is an FC-group if and only if it is locally (soluble-by-finite) and has the property K

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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

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We shall say that a group G has the property N 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

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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 X g | is finite for each element g of G X is said to be strongly inert in G if the index | X,X g : X| is finite for each element g of G

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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

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The inequality |X:X X g |≤ | X,X g : X g | 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

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Clearly, any normal subgroup of an arbitrary group is strong inert and so inert On the other hand, finite subgroups are inert but in general they are not strongly inert In fact the infinite dihedral group is inertial but it is not strongly inertial Note also that Tarski groups are inertial

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Theorem 12 (D.J.S. Robinson, 2006) Let G be a finitely generated soluble-by-finite group. Then G is inertial if and only if it has an abelian normal subgroup A of finite index such that every element of G induces on A a power automorphism In the same paper Robinson also provides a complete classification of soluble-by-finite minimax groups which are inertial

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A special class of strongly inertial groups: groups in which every subgroup has finite index in its normal closure Theorem 13 (B.H. Neumann, 1955) In a group G every subgroup has finite index in its nrmal closure if and only if the commutator subgroup G’ of G is finite

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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

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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

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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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