slides de giovanni fc groups ischia group theory

slides de giovanni fc groups ischia group theory

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

2. A groupG iscalledanFC-groupifevery
elementofG hasonlyfinitelymany
conjugates, or equivalentlyifthe index
|G:CG(x)|isfinite foreachelementx
Finite groupsand abeliangroupsare
obviouslyexamplesofFC-groups
Anydirectproductoffinite or abelian
subgroupshasthe propertyFC

3. FC-groupshavebeenintroduced70 years
ago, and relevantcontributionshavebeen
givenbyseveralimportantauthors
R. Baer, P. Hall, B.H.Neumann, Y.M.Gorcakov,
M.J.Tomkinson, L.A.Kurdachenko
… and manyothers

4. Clearlygroupswhosecentrehasfinite index
are FC-groups
IfGisa groupand xisanyelementofG,
theconjugacyclassofx iscontained
in the cosetxG’
ThereforeifG’ isfinite, the groupG has
boundedlyfinite conjugacyclasses

5. Theorem1 (B.H.Neumann, 1954)
A groupG hasboundedlyfinite
conjugacyclassesifand onlyif
itscommutatorsubgroupG’ isfinite

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 propertiesCand C∞
A groupG hasthe propertyCifthe set
{X’ | X ≤ G} isfinite
A groupG hasthe propertyC∞ifthe set
{X’ | X ≤ G, X infinite} isfinite

13. Tarskigroups(i.e. infinite simplegroupswhose
propernon-trivialsubgroupshaveprime order)
haveobviouslythe propertyC
A groupG islocallygradedifeveryfinitely
generatednon-trivialsubgroupofGcontainsa
propersubgroupoffinite index
Alllocally(soluble-by-finite) groupsare locally
graded

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 Gbe a Cernikov group, and let J be its
finite residual
(i.e. the largest divisible abelian subgroup of G).
We say that G is irreducibleif [J,G]≠{1} and J has no
infinite proper K-invariant subgroups for
CG(J)<K≤G

16. Theorem7 (F. de Giovanni –D.J.S.Robinson, 2005)
LetG bea locallygradedgroupwiththe
propertyC∞. TheneitherG’ isfinite or G isan
irreducibleCernikovgroup

17. Recall that a group Gis called metahamiltonianif
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, Theorem6 can beprovedalsoifthe
conditionCisimposedonlytonon-normal
subgroups
Theorem8 (F. De Mari –F. de Giovanni, 2006)
LetG bea locallygradedgroupwithfinitelymany
derivedsubgroupsofnon-normalsubgroups. Then
the commutatorsubgroupG’ ofG isfinite
A similarremarkholdsalsoforthe propertyC∞

19. The properties Kand K∞
A group G has the property Kif for each element
xof Gthe set
{[x,H]| H ≤G} is finite
A group G has the property K∞if for each element
xof Gthe 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. Theorem9 (M. De Falco –F. de Giovanni –C. Musella, 2010)
A groupG isanFC-groupifand onlyifitislocally
(soluble-by-finite) and hasthe 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 Ghas the propertyN if
for each subgroup X of Gthe 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 Xbe a subgroup of G.
Xis said to be inertin G if the index |X:XÇXg|is finite for each element g of G
X is said to be strongly inertin Gif the index |áX,Xgñ:X| is finite for each element g of G

25. A group Gis called inertialif all its
subgroups are inert
Similarly, Gis strongly inertialif every
subgroup of Gis strongly inert

26. The inequality
|X:XÇXg|≤ |áX,Xgñ: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, anynormalsubgroupofanarbitrary
groupisstrong inertand so inert
On the otherhand, finite subgroupsare inertbutin
generaltheyare notstronglyinert
In factthe infinite dihedralgroupisinertial
butitisnotstronglyinertial
Note alsothatTarskigroupsare inertial

28. Theorem12 (D.J.S.Robinson, 2006)
LetG bea finitelygeneratedsoluble-by-finitegroup.
ThenG isinertialifand onlyifithasanabelian
normalsubgroupA offinite indexsuchthatevery
elementofG induceson A apowerautomorphism
In the samepaperRobinson alsoprovidesa
complete classificationofsoluble-by-finite
minimax groupswhichare inertial

29. A specialclassofstronglyinertialgroups:
groupsin whicheverysubgrouphasfinite index
in itsnormalclosure
Theorem13 (B.H.Neumann, 1955)
In a groupG everysubgrouphasfinite indexin its
nrmalclosureifand onlyifthe commutatorsubgroup
G’ ofG isfinite

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