The intersection map of subgroups and the classes $${\mathcal {PST}_c}$$ , $${\mathcal {PT}_c}$$ and $${\mathcal {T}_c}$$

Abstract

A group G is called a \({\mathcal {T}_{c}}\)-group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes \({\mathcal {PT}_{c}}\) and \({\mathcal {PST}_{c}}\) are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that \({X=Y \cap H}\). We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of \({\mathcal {T}_c}\), \({\mathcal {PT}_c}\) and \({\mathcal {PST}_c}\) groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups.