Uncountable groups with restrictions on subgroups of large cardinality. (English) Zbl 1333.20037
The authors look for group classes \(\mathbf P\) and (possibly) additional conditions for groups \(G\) of cardinality \(\aleph\) such that \(G\) is a \(\mathbf P\)-group whenever all proper equicardinal subgroups are \(\mathbf P\)-groups. Here with \(\aleph\) is meant an uncountable cardinality. If \(G\) has no simple homomorphic images of cardinality \(\aleph\) and \(\aleph\) is regular, \(G\) is an FC-group if all equicardinal proper subgroups are (Theorem 3.2). If the locally graded group \(G\) has no simple homomorphic images of cardinality \(\aleph\) and \(\aleph\) is regular, \(G\) is nilpotent-by-finite if all equicardinal subgroups are (Theorem 4.6). The absence of equicardinal simple homomorphic images is needed because of the existence of groups with all proper subgroups of strictly lower cardinality (Jónsson groups).
Reviewer: Hermann Heineken (Würzburg)
MSC:
| 20F24 | FC-groups and their generalizations |
| 03E75 | Applications of set theory |
| 20E07 | Subgroup theorems; subgroup growth |
| 20F19 | Generalizations of solvable and nilpotent groups |
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