The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\\aleph$ in which all proper subgroups of cardinality $\\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\\aleph$ has cofinality strictly larger than $\\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

Nilpotency in uncountable groups

A class Open image in new window of groups is said to be countably recognizable if a group belongs to Open image in new window whenever all its countable subgroups lie in Open image in new window. It is proved here that most of the relevant group classes of generalized nilpotent groups are countably recognizable. Moreover, some further countably recognizable group classes are exhibited.

Countable recognizability and nilpotency properties of groups

https://msp.org/pjm/2021/312-2/pjm-v312-n2-p08-s.pdf

Cohopfian groups and accessible group classes

A subgroup X of a group G is called permutable if $$XY=YX$$
 for all subgroups Y of G. It is well known that permutable subgroups play an important role in many problems in group theory, and the purpose of this paper is to describe the structure of uncountable groups of regular cardinality 
 $$\aleph $$
 in which all large subgroups are permutable.

/pdf/permutability-in-uncountable-groups-2wn2xhvm2f.pdf

Permutability in uncountable groups

Abstract A subgroup H of a group G is pronormal in G if each of its conjugates \n$H^g$\n in G is conjugate to it in the subgroup \n$\\langle H,H^g\\rangle $\n ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

/pdf/groups-with-many-pronormal-subgroups-4snh7de5p2.pdf

Groups with many pronormal subgroups

If &#x1d51b; is a class of groups, a group G is minimal non-&#x1d51b; if it is not an &#x1d51b;-group, but all its proper subgroups belong to &#x1d51b;. The aim of this paper is to prove that for an infinite locally graded group, the property of being minimal non-hypercentral and that of being minimal non-hypercyclic are equivalent. Moreover, the main properties of infinite minimal non-hypercentral groups are described. In the last section, we study groups of infinite rank in which all proper subgroups of infinite rank satisfy a generalized supersolubility condition.

Marco Trombetti

Papers

Infinite minimal non-hypercyclic groups

Uncountable groups with restrictions on subgroups of large cardinality

Nilpotency in uncountable groups

Countable recognizability and nilpotency properties of groups

Cohopfian groups and accessible group classes