Groups with Finiteness Conditions

on Conjugates and Commutators

Francesco de GiovanniUniversità 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:CG(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

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

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

Theorem 1 (B.H. Neumann, 1954)

A group G has boundedly finite conjugacy classes if and only if its commutator subgroup G’

is finite

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

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

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

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

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

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

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

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

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

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

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

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

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∞

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

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

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

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

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

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

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

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

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

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

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

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

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

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