Double chain conditions in infinite groups Mattia Brescia - - PowerPoint PPT Presentation

Double chain conditions in infinite groups

Mattia Brescia

Universit` a degli Studi di Napoli Federico II

Gruppen und topologische Gruppen 2017

Trento – 16 June, 2017

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Finiteness Conditions

Let U be the universe of all groups and let F be the class of finite

• groups. Any intermediate class between them, i.e. any X such

that F X U, is said to be a finiteness class. The property of belonging to such class is called a finiteness or finitary condition. Classical non-trivial examples of finiteness conditions are Locally finiteness; Periodicity; Hopficity; ...

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Finiteness Conditions

Let U be the universe of all groups and let F be the class of finite

• groups. Any intermediate class between them, i.e. any X such

that F X U, is said to be a finiteness class. The property of belonging to such class is called a finiteness or finitary condition. Classical non-trivial examples of finiteness conditions are Locally finiteness; Periodicity; Hopficity; ...

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Chain Conditions

Let T be a totally ordered set and χ be a group theoretical prop- erty. We will say that a group G satisfies the T-chain condition on χ-subgroups if there is no increasing function between T and the set of the χ-subgroups of G ordered by inclusion. Such class will be called a chain class and will here be denoted by C(T, χ). Very known examples of chain classes are For T = (N, <) and χ =”to be a group”, C(T; χ) = Max; For T = (N, >) and χ =”to be a group”, C(T; χ) = Min; For T = (N, <) and χ =”to be a finite group”, C(T; χ) = Max-f ...

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Chain Conditions

Let T be a totally ordered set and χ be a group theoretical prop- erty. We will say that a group G satisfies the T-chain condition on χ-subgroups if there is no increasing function between T and the set of the χ-subgroups of G ordered by inclusion. Such class will be called a chain class and will here be denoted by C(T, χ). Very known examples of chain classes are For T = (N, <) and χ =”to be a group”, C(T; χ) = Max; For T = (N, >) and χ =”to be a group”, C(T; χ) = Min; For T = (N, <) and χ =”to be a finite group”, C(T; χ) = Max-f ...

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Some history

First examples coming in were Noetherian groups (R. Baer, K. A. Hirsch, O. H. Kegel, ...); Artinian groups (S. N. ˇ Cernikov, V. P. ˇ Sunkov, D. I. Zaicev, ...); Max-n groups (P. Hall, D. H. McLain, J. S. Wilson, ...); Max-ab groups (A. I. Malˇ cev, B. I. Plotkin, O. J. Schmidt, ...).

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Aiming for cool results

Some early relevant questions were Is Max = (PC)F? Is Min = ˇ C? Is Max-sn = Max in the universe of locally soluble groups?

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Ending up with monsters

The answers to which are Is Max = (PC)F? NO. (Thank you, Ol’ˇ sanski˘ i) Is Min = ˇ C? NO. (Thank you again, Ol’ˇ sanski˘ i) Is Max-sn = Max in the universe of locally soluble groups? Who knows!

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Ending up with monsters

The answers to which are Is Max = (PC)F? NO. (Thank you, Ol’ˇ sanski˘ i) Is Min = ˇ C? NO. (Thank you again, Ol’ˇ sanski˘ i) Is Max-sn = Max in the universe of locally soluble groups? Who knows!

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Ending up with monsters

The answers to which are Is Max = (PC)F? NO. (Thank you, Ol’ˇ sanski˘ i) Is Min = ˇ C? NO. (Thank you again, Ol’ˇ sanski˘ i) Is Max-sn = Max in the universe of locally soluble groups? Who knows!

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Changing the point of view

For many years the quasi-totality of investigations about chain conditions were about changing the group theoretical property χ and showing results about Max-χ or Min-χ, i.e. about the classes

• f the groups satisfying the maximal or the minimal condition
• n χ-subgroups.

A recent inspiration came reading the works of D. I. Zaicev and T.

• S. Shores who, in particular, studied the class of C(Z; U), where

U is the trivial universal property.

• D. I. Zaicev (1971), T. S. Shores (1973) - Let G be a locally radical
• group. Then G satisfies the double chain condition on subgroups

iff it satisfies either the maximal or the minimal condition on subgroups.

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Changing the point of view

For many years the quasi-totality of investigations about chain conditions were about changing the group theoretical property χ and showing results about Max-χ or Min-χ, i.e. about the classes

• f the groups satisfying the maximal or the minimal condition
• n χ-subgroups.

A recent inspiration came reading the works of D. I. Zaicev and T.

• S. Shores who, in particular, studied the class of C(Z; U), where

U is the trivial universal property.

• D. I. Zaicev (1971), T. S. Shores (1973) - Let G be a locally radical
• group. Then G satisfies the double chain condition on subgroups

iff it satisfies either the maximal or the minimal condition on subgroups.

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Changing the point of view

So in 2005 there came the first new work on the so-called ”Double chain condition”.

• F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property
• f ”being normal”.

Then, in the universe of residually finite groups, DCn = Max-n.

• F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property
• f ”being normal”. Then, in the universe of periodic soluble

groups, DCn = Min-n.

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Changing the point of view

So in 2005 there came the first new work on the so-called ”Double chain condition”.

• F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property
• f ”being normal”.

Then, in the universe of residually finite groups, DCn = Max-n.

• F. De Mari, F. de Giovanni (2005) - Let n be the subgroup property
• f ”being normal”. Then, in the universe of periodic soluble

groups, DCn = Min-n.

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Changing the point of view

Moreover in the same paper

• F. De Mari, F. de Giovanni (2005) - Let nn be the subgroup

property of ”being not normal”. Then, in the universe of locally radical groups, G is a DCnn-group if and only if G satisfies either Max-nn or Min-nn. So is everything this predictable?

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Changing the point of view

Moreover in the same paper

• F. De Mari, F. de Giovanni (2005) - Let nn be the subgroup

property of ”being not normal”. Then, in the universe of locally radical groups, G is a DCnn-group if and only if G satisfies either Max-nn or Min-nn. So is everything this predictable?

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Going on with the ordering type of Z

Theorem

• F. de Giovanni, M. B. – 2015

Let G be a radical group. G satisfies DCsn if and only if G satisfies one of the following: G satisfies Max-sn; G satisfies Min-sn; G = HJ where J is the finite residual of G, H is polycyclic, CH(J) is finite and every subnormal subgroup of G is either Min or properly contains J. Not everything is this predictable, indeed!

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Going on with the ordering type of Z

Theorem

• F. de Giovanni, M. B. – 2015

Let G be a radical group. G satisfies DCsn if and only if G satisfies one of the following: G satisfies Max-sn; G satisfies Min-sn; G = HJ where J is the finite residual of G, H is polycyclic, CH(J) is finite and every subnormal subgroup of G is either Min or properly contains J. Not everything is this predictable, indeed!

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Going on with the ordering type of Z

Theorem

• F. de Giovanni, M. B. – 2015

Let G be a radical group. G satisfies DCsn if and only if G satisfies one of the following: G satisfies Max-sn; G satisfies Min-sn; G = HJ where J is the finite residual of G, H is polycyclic, CH(J) is finite and every subnormal subgroup of G is either Min or properly contains J. Not everything is this predictable, indeed!

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Double chains vs non-subnormal subgroups

M.B. (2016) - Let sn be the subgroup property of ”being not sub- normal”. Then, in the universe of infinite locally finite groups, DCsn = ˇ C ∪ S, where S is the class of groups with every sub- group subnormal.

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Double chains vs snn-subgroups

M.B. (2016) - Let G be a finitely generated soluble DCsnn-group. Then G satisfies Max. M.B. (2016) - Let G be a Baer DCsnn-group. Then G is nilpotent. In particular, G satisfies either Max-nn or Min-nn.

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Double chains vs snn-subgroups

M.B. (2016) - Let G be a finitely generated soluble DCsnn-group. Then G satisfies Max. M.B. (2016) - Let G be a Baer DCsnn-group. Then G is nilpotent. In particular, G satisfies either Max-nn or Min-nn.

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Double chains vs snn-subgroups

M.B. (2016) - Let G be a periodic soluble DCsnn-group. Then G either satisfies Min or G is abelian, provided that G is a p-group for an odd prime p; G contains a finite number of subnormal non-normal subgroups, provided that G is a 2-group. M.B. (2016) - Let G be a soluble DCsnn-group with non-periodic Fitting subgroup. Then either G is minimax or every subnormal subgroup of G has finite index in G. In particular, if G is not minimax, G satisfies Max-snn.

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Double chains vs snn-subgroups

M.B. (2016) - Let G be a periodic soluble DCsnn-group. Then G either satisfies Min or G is abelian, provided that G is a p-group for an odd prime p; G contains a finite number of subnormal non-normal subgroups, provided that G is a 2-group. M.B. (2016) - Let G be a soluble DCsnn-group with non-periodic Fitting subgroup. Then either G is minimax or every subnormal subgroup of G has finite index in G. In particular, if G is not minimax, G satisfies Max-snn.

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Double chains vs non-pronormal subgroups

M.B. (2016) - Let G be a finitely generated soluble DCnp-group. Then G satisfies Max. M.B. (2016) - Let G be a periodic locally radical DCnp-group. Then either G satisfies Min or every subgroup of G is pronormal.

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Double chains vs non-pronormal subgroups

M.B. (2016) - Let G be a finitely generated soluble DCnp-group. Then G satisfies Max. M.B. (2016) - Let G be a periodic locally radical DCnp-group. Then either G satisfies Min or every subgroup of G is pronormal.

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Double chains vs non-pronormal subgroups

M.B. (2016) - Let G be a locally nilpotent DCnp-group. Then G either satisfies Min or is nilpotent. In particular, G satisfies either Max-np or Min-np. M.B. (2016) - Let G be a radical DCnp-group. Then G is either minimax or a T-group.

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Double chains vs non-pronormal subgroups

M.B. (2016) - Let G be a locally nilpotent DCnp-group. Then G either satisfies Min or is nilpotent. In particular, G satisfies either Max-np or Min-np. M.B. (2016) - Let G be a radical DCnp-group. Then G is either minimax or a T-group.

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Chain conditions - (not?) a conclusion

Thank you for your attention

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