Almost Hamiltonian groups. (Q817159)

The following problem is investigated: which properties \(P\) transfer (or do not transfer) from all cyclic subgroups, or all Abelian subgroups, to all arbitrary subgroups? The following assertion is immediate: Let \(P\) be a subgroup property closed with respect to joins of cyclic subgroups. Then, the following statements are equivalent: (i) \(G\) is a \(P\)-Hamiltonian group (i.e., every subgroup of \(G\) has property \(P\)), (ii) All Abelian subgroups of \(G\) have property \(P\), (iii) All cyclic subgroups of \(G\) have property \(P\). The authors use the following notations: \(G\) is an \(AH1\) group if every subgroup of \(G\) has finite index in a normal subgroup, \(G\) is an \(AH2\) group if every Abelian subgroup of \(G\) has finite index in a normal subgroup, \(G\) is an \(AH3\) group if every cyclic subgroup of \(G\) has finite index in a normal subgroup, \([X]\) is the class of groups having property \(X\). The main result of the paper yields that the following relations are true: (i) \([AH1]=[AH2]\), (ii) \([AH3]=[FC]\), (iii) \([AH1]\) is a proper subclass of \([AH3]\).