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Groups whose non-Normal Subgroups are
Metahamiltonian
Dario Esposito
Università degli Studi di Napoli Federico II
Young Researchers in Algebra Conference - 2019
Napoli � September, 18th
Dario Esposito
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Metahamiltonian groups
In 1966, G.M. Romalis and N.F. Sesekin introducedmetahamiltonian groups.
They say a group G is metahamiltonian if:
∀H ≤ G , H 6∈ A⇒ H / G
where A is the class of abelian groups.
We will use H to denote the class of metahamiltonian groups,which is trivially SH-closed, but not closed by extensions or evendirect products.
Dario Esposito
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Metahamiltonian groups
In 1966, G.M. Romalis and N.F. Sesekin introducedmetahamiltonian groups.
They say a group G is metahamiltonian if:
∀H ≤ G , H 6∈ A⇒ H / G
where A is the class of abelian groups.
We will use H to denote the class of metahamiltonian groups,which is trivially SH-closed, but not closed by extensions or evendirect products.
Dario Esposito
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Metahamiltonian groups
In 1966, G.M. Romalis and N.F. Sesekin introducedmetahamiltonian groups.
They say a group G is metahamiltonian if:
∀H ≤ G , H 6∈ A⇒ H / G
where A is the class of abelian groups.
We will use H to denote the class of metahamiltonian groups,which is trivially SH-closed, but not closed by extensions or evendirect products.
Dario Esposito
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Metahamiltonian groups
Restricting out attention to locally soluble groups to exclude Tarskigroups, they proved
Theorem 1. [G.M. Romalis - N.F. Sesekin, 1966]
Let G be a locally soluble metahamiltonian group. Then G issoluble, the derived length of G is at most 3 and the commutatorsubgroup G ′ is �nite with prime-power order.
A simpli�ed proof of the result in the larger class of locally gradedgroups can be found in a paper by F. De Mari and F. de Giovanni(2006) about groups with few normalizers of non-abeliansubgroups.A group G is locally graded if all of its non-trivial �nitely generatedsubgroup have proper subgroups of �nite index.
Dario Esposito
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Metahamiltonian groups
Restricting out attention to locally soluble groups to exclude Tarskigroups, they proved
Theorem 1. [G.M. Romalis - N.F. Sesekin, 1966]
Let G be a locally soluble metahamiltonian group. Then G issoluble, the derived length of G is at most 3 and the commutatorsubgroup G ′ is �nite with prime-power order.
A simpli�ed proof of the result in the larger class of locally gradedgroups can be found in a paper by F. De Mari and F. de Giovanni(2006) about groups with few normalizers of non-abeliansubgroups.A group G is locally graded if all of its non-trivial �nitely generatedsubgroup have proper subgroups of �nite index.
Dario Esposito
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Metahamiltonian groups
Restricting out attention to locally soluble groups to exclude Tarskigroups, they proved
Theorem 1. [G.M. Romalis - N.F. Sesekin, 1966]
Let G be a locally soluble metahamiltonian group. Then G issoluble, the derived length of G is at most 3 and the commutatorsubgroup G ′ is �nite with prime-power order.
A simpli�ed proof of the result in the larger class of locally gradedgroups can be found in a paper by F. De Mari and F. de Giovanni(2006) about groups with few normalizers of non-abeliansubgroups.A group G is locally graded if all of its non-trivial �nitely generatedsubgroup have proper subgroups of �nite index.
Dario Esposito
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Metahamiltonian groups
Restricting out attention to locally soluble groups to exclude Tarskigroups, they proved
Theorem 1. [G.M. Romalis - N.F. Sesekin, 1966]
Let G be a locally soluble metahamiltonian group. Then G issoluble, the derived length of G is at most 3 and the commutatorsubgroup G ′ is �nite with prime-power order.
A simpli�ed proof of the result in the larger class of locally gradedgroups can be found in a paper by F. De Mari and F. de Giovanni(2006) about groups with few normalizers of non-abeliansubgroups.A group G is locally graded if all of its non-trivial �nitely generatedsubgroup have proper subgroups of �nite index.
Dario Esposito
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The class H can also be shown to be L-closed, i.e.:
Proposition
Let G be a group whose �nitely generated subgroups aremetahamiltonian. Then G itself is metahamiltonian.
Dario Esposito
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The class H can also be shown to be L-closed, i.e.:
Proposition
Let G be a group whose �nitely generated subgroups aremetahamiltonian. Then G itself is metahamiltonian.
Dario Esposito
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Generalized metahamiltonian groups
Some natural extensions of the class of metahamiltonian groups arestudied in a paper by M. De Falco, F. de Giovanni and C. Musellain 2009.
Theorem 2. [M. De Falco - F. de Giovanni - C. Musella, 2009]
Let G be a locally graded minimal-non-H group. Then G is �nite.
Of course such a group is either perfect (one example is A5) orsoluble of derived length at most 4.
Dario Esposito
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Generalized metahamiltonian groups
Some natural extensions of the class of metahamiltonian groups arestudied in a paper by M. De Falco, F. de Giovanni and C. Musellain 2009.
Theorem 2. [M. De Falco - F. de Giovanni - C. Musella, 2009]
Let G be a locally graded minimal-non-H group. Then G is �nite.
Of course such a group is either perfect (one example is A5) orsoluble of derived length at most 4.
Dario Esposito
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Generalized metahamiltonian groups
They also proved
Theorem 3. [M. De Falco - F. de Giovanni - C. Musella, 2009]
Let G be a �nitely generated hyper(abelian-or-�nite) group whose�nite homomorphic images are in H. Then G is metahamiltonian.
and
Theorem 4. [M. De Falco, F. de Giovanni, C. Musella, 2009]
Let G be a locally graded group satisfying the minimal condition onnon-metahamiltonian subgroups. Then G is either �ernikov ormetahamiltonian.
Dario Esposito
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Generalized metahamiltonian groups
They also proved
Theorem 3. [M. De Falco - F. de Giovanni - C. Musella, 2009]
Let G be a �nitely generated hyper(abelian-or-�nite) group whose�nite homomorphic images are in H. Then G is metahamiltonian.
and
Theorem 4. [M. De Falco, F. de Giovanni, C. Musella, 2009]
Let G be a locally graded group satisfying the minimal condition onnon-metahamiltonian subgroups. Then G is either �ernikov ormetahamiltonian.
Dario Esposito
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k-metahamiltonian groups
LetH0 = A
be the class of all abelian groups.We de�ne, recursively, the class Hk in the following way.A group G is in the class Hk if and only if
∀H ≤ G , H 6∈ Hk−1 ⇒ H / G
and refer to groups in the class Hk as k-metahamiltonian groups.With k = 1 we have the usual metahamiltonian groups.Obviously we have:
H0 ⊆ H1 ⊆ H2 ⊆ . . .
Dario Esposito
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k-metahamiltonian groups
LetH0 = A
be the class of all abelian groups.We de�ne, recursively, the class Hk in the following way.A group G is in the class Hk if and only if
∀H ≤ G , H 6∈ Hk−1 ⇒ H / G
and refer to groups in the class Hk as k-metahamiltonian groups.With k = 1 we have the usual metahamiltonian groups.Obviously we have:
H0 ⊆ H1 ⊆ H2 ⊆ . . .
Dario Esposito
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Properties of k-metahamiltonian groups
Theorem 5. [F. de Giovanni, D.E., M. Trombetti 2019]
Let G be a locally graded Hk -group. Then the commutatorsubgroup G ′ is �nite and if G is soluble, its derived length does notexceed 3k .
Many other interesting properties of metahamiltonian groups canbe proved also for Hk -groups:
There exist non soluble H2-groups, e.g. A5, and H2-groupswhose commutator subgroup is not of prime-power order, e.g.GL(2, 3) whose commutator subgroup is SL(2, 3) which is agroup of order 24 = 23 · 3.Every locally graded minimal-non-Hk group is �nite.
Dario Esposito
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Properties of k-metahamiltonian groups
Theorem 5. [F. de Giovanni, D.E., M. Trombetti 2019]
Let G be a locally graded Hk -group. Then the commutatorsubgroup G ′ is �nite and if G is soluble, its derived length does notexceed 3k .
Many other interesting properties of metahamiltonian groups canbe proved also for Hk -groups:
There exist non soluble H2-groups, e.g. A5, and H2-groupswhose commutator subgroup is not of prime-power order, e.g.GL(2, 3) whose commutator subgroup is SL(2, 3) which is agroup of order 24 = 23 · 3.Every locally graded minimal-non-Hk group is �nite.
Dario Esposito
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Properties of k-metahamiltonian groups
Theorem 5. [F. de Giovanni, D.E., M. Trombetti 2019]
Let G be a locally graded Hk -group. Then the commutatorsubgroup G ′ is �nite and if G is soluble, its derived length does notexceed 3k .
Many other interesting properties of metahamiltonian groups canbe proved also for Hk -groups:
There exist non soluble H2-groups, e.g. A5, and H2-groupswhose commutator subgroup is not of prime-power order, e.g.GL(2, 3) whose commutator subgroup is SL(2, 3) which is agroup of order 24 = 23 · 3.Every locally graded minimal-non-Hk group is �nite.
Dario Esposito
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Properties of k-metahamiltonian groups
The class Hk is a local class.
Finitely generated Hk -groups are polycyclic-by-�nite.
Locally graded groups satisfying the minimal condition onnon-k-metahamiltonian subgroups are necessarily �ernikov ork-metahamiltonian.
Polycyclic-by-�nite groups whose �nite homomorphic imagesall lie in the class Hk are Hk .
Dario Esposito
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Properties of k-metahamiltonian groups
The class Hk is a local class.
Finitely generated Hk -groups are polycyclic-by-�nite.
Locally graded groups satisfying the minimal condition onnon-k-metahamiltonian subgroups are necessarily �ernikov ork-metahamiltonian.
Polycyclic-by-�nite groups whose �nite homomorphic imagesall lie in the class Hk are Hk .
Dario Esposito
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Properties of k-metahamiltonian groups
The class Hk is a local class.
Finitely generated Hk -groups are polycyclic-by-�nite.
Locally graded groups satisfying the minimal condition onnon-k-metahamiltonian subgroups are necessarily �ernikov ork-metahamiltonian.
Polycyclic-by-�nite groups whose �nite homomorphic imagesall lie in the class Hk are Hk .
Dario Esposito
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Properties of k-metahamiltonian groups
The class Hk is a local class.
Finitely generated Hk -groups are polycyclic-by-�nite.
Locally graded groups satisfying the minimal condition onnon-k-metahamiltonian subgroups are necessarily �ernikov ork-metahamiltonian.
Polycyclic-by-�nite groups whose �nite homomorphic imagesall lie in the class Hk are Hk .
Dario Esposito
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Groups with �nitely many normalizers of
non-k-metahamiltonian subgroups
F. De Mari and F. de Giovanni proved the following resultconcerning groups whose non-abelian subgroups have �nitely manynormalizers.
Theorem 6. [F. de Giovanni - F. De Mari, 2006]
Let G be a locally graded group with �nitely many normalizers ofnon-abelian subgroups. Then G has �nite commutator subgroup.
This can be extended to
Theorem 7. [F. de Giovanni, D.E., M. Trombetti 2019]
Let G be a locally graded group with �nitely many normalizers ofsubgroups which are not Hk . Then G has �nite commutatorsubgroup.
Dario Esposito
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Groups with �nitely many normalizers of
non-k-metahamiltonian subgroups
F. De Mari and F. de Giovanni proved the following resultconcerning groups whose non-abelian subgroups have �nitely manynormalizers.
Theorem 6. [F. de Giovanni - F. De Mari, 2006]
Let G be a locally graded group with �nitely many normalizers ofnon-abelian subgroups. Then G has �nite commutator subgroup.
This can be extended to
Theorem 7. [F. de Giovanni, D.E., M. Trombetti 2019]
Let G be a locally graded group with �nitely many normalizers ofsubgroups which are not Hk . Then G has �nite commutatorsubgroup.
Dario Esposito
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∞-metahamiltonian groups
One can also de�ne the class
H∞ =⋃k∈N
Hk .
Notice now that if G is in this class, G cannot have any in�nitedescending chain of subgroups:
G = G0 > G1 > G2 > . . . with Gi+1 6 Gi
As a matter of fact, one can show that H∞ is precisely the class ofgroups with a bound on the length of such "bad" chains.Notice that we already know that
H∞ ⊆ FA
i.e. ∞-metahamiltonian groups have �nite commutator subgroup.Dario Esposito
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∞-metahamiltonian groups
One can also de�ne the class
H∞ =⋃k∈N
Hk .
Notice now that if G is in this class, G cannot have any in�nitedescending chain of subgroups:
G = G0 > G1 > G2 > . . . with Gi+1 6 Gi
As a matter of fact, one can show that H∞ is precisely the class ofgroups with a bound on the length of such "bad" chains.Notice that we already know that
H∞ ⊆ FA
i.e. ∞-metahamiltonian groups have �nite commutator subgroup.Dario Esposito
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∞-metahamiltonian groups
We can also show the converse (i.e. FA ⊆ H∞, hence the equalityof the classes).
For if G is a group such that |G ′| = n = pn11 . . . pnll , then anydescending chain {Gi} of subgroups of G related by Gi+1 6 Gi isbounded in length by f (n) = n1 + n2 + . . .+ nl and so G is Hf (n).
The fact that a group with bound f (n) on the length of this type ofchains is Hf (n) can be proved by induction on n1 + . . .+ nl . Thebasis of induction is the observation that groups with commutatorof prime order are necessarily metahamiltonian.
Dario Esposito
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∞-metahamiltonian groups
We can also show the converse (i.e. FA ⊆ H∞, hence the equalityof the classes).
For if G is a group such that |G ′| = n = pn11 . . . pnll , then anydescending chain {Gi} of subgroups of G related by Gi+1 6 Gi isbounded in length by f (n) = n1 + n2 + . . .+ nl and so G is Hf (n).
The fact that a group with bound f (n) on the length of this type ofchains is Hf (n) can be proved by induction on n1 + . . .+ nl . Thebasis of induction is the observation that groups with commutatorof prime order are necessarily metahamiltonian.
Dario Esposito
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∞-metahamiltonian groups
We can also show the converse (i.e. FA ⊆ H∞, hence the equalityof the classes).
For if G is a group such that |G ′| = n = pn11 . . . pnll , then anydescending chain {Gi} of subgroups of G related by Gi+1 6 Gi isbounded in length by f (n) = n1 + n2 + . . .+ nl and so G is Hf (n).
The fact that a group with bound f (n) on the length of this type ofchains is Hf (n) can be proved by induction on n1 + . . .+ nl . Thebasis of induction is the observation that groups with commutatorof prime order are necessarily metahamiltonian.
Dario Esposito
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Groups whose non-normal subgroups belong to a class X
Actually, we can start from any SH-closed class of groups X andde�ne:
X0 = X
and, for any k ∈ N, say that a group G is in the class Xk if andonly if:
∀H ≤ G , H 6∈ Xk−1 ⇒ H / G .
We can then also de�ne the class X∞ as⋃k∈N
Xk .
Dario Esposito
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Groups whose non-normal subgroups belong to a class X
If one assumes that X satis�es the following properties
SHL-closed
X ⊆ FA
Minimal-non-X groups are �nite
Polycyclic-by-�nite groups whose �nite homomorphic imagesare X are also X
Then all properties in the list also hold for Xk for any k ∈ N
Dario Esposito
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Groups whose non-normal subgroups belong to a class X
If one assumes that X satis�es the following properties
SHL-closed
X ⊆ FA
Minimal-non-X groups are �nite
Polycyclic-by-�nite groups whose �nite homomorphic imagesare X are also X
Then all properties in the list also hold for Xk for any k ∈ N
Dario Esposito
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Groups whose non-normal subgroups belong to a class X
If one assumes that X satis�es the following properties
SHL-closed
X ⊆ FA
Minimal-non-X groups are �nite
Polycyclic-by-�nite groups whose �nite homomorphic imagesare X are also X
Then all properties in the list also hold for Xk for any k ∈ N
Dario Esposito
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Thank you for the attention!!!
Dario Esposito