slides de giovanni fc groups ischia group theory

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

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