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Groups with nearly modular subgroup lattice

100%
de Giovanni F., Musella C.
Colloquium Mathematicum
|
2001
|
tom 88
|
nr 1
13-20
EN
A subgroup H of a group G is nearly normal if it has finite index in its normal closure $H^{G}$. A relevant theorem of B. H. Neumann states that groups in which every subgroup is nearly normal are precisely those with finite commutator subgroup. We shall say that a subgroup H of a group G is nearly modular if H has finite index in a modular element of the lattice of subgroups of G. Thus nearly modular subgroups are the natural lattice-theoretic translation of nearly normal subgroups. In this article we study the structure of groups in which all subgroups are nearly modular, proving in particular that a locally graded group with this property has locally finite commutator subgroup.
2
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A note on groups with few isomorphism classes of subgroups

100%
de Giovanni F., Russo A.
Colloquium Mathematicum
|
2016
|
tom 144
|
nr 2
265-271
EN
The structure of infinite groups in which any two (proper) subgroups of the same cardinality are isomorphic is described within the universe of locally graded groups. The corresponding problem for finite groups was considered by R. Armstrong (1958).
3
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A note on groups of infinite rank whose proper subgroups are abelian-by-finite

100%
de Giovanni F., Saccomanno F.
Colloquium Mathematicum
|
2014
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tom 137
|
nr 2
165-170
EN
It is proved that if G is a locally (soluble-by-finite) group of infinite rank in which every proper subgroup of infinite rank contains an abelian subgroup of finite index, then all proper subgroups of G are abelian-by-finite.
4
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Groups satisfying the maximal condition on subnormal non-normal subgroups

100%
De Mari F., de Giovanni F.
Colloquium Mathematicum
|
2005
|
tom 103
|
nr 1
85-98
EN
The structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the maximal condition is described, taking as a model the known theory of groups in which normality is a transitive relation.
5
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Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank

81%
De Falco M., de Giovanni F., Musella C.
Colloquium Mathematicum
|
2013
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tom 131
|
nr 2
233-239
EN
It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.
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