On a class of groups with many quasinormal subgroups (Q1078665)

The author studies the structure of groups G of the following class: If \(\{H_ i\}_{i\in I}\) is a family of subgroups of G such that \(G=<H_ i:\) \(i\in I>\) then for each i,j\(\in I\), any subgroup contained in \(H_ i\vee H_ j\) is quasinormal in G. It is known that a periodic group with this property is metabelian. The author improves the mentioned result in the following way: If \(\sigma\) is a projectivity of the group G into the group \(\bar G\) and \(N\triangleleft G\), then \((N^{\sigma}[N^{\sigma},2\bar G])^{\sigma -1}/N\) is a modular group.