Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

ABSTRACT

A group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(𝔓𝔉)𝔄) if, and only if, it is non-perfect minimal non-FC, where (𝔓𝔉)𝔄 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(𝔓𝔉)𝔄) is itself in ((PF)L∪(𝔓𝔉)𝔄) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(𝔓𝔉)𝔄)-subgroups.

KEYWORDS:

• Conjugacy classes
• layers
• (PF)C-groups
• (PF)L-groups
• Prüfer rank

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F24
• 20E45

Acknowledgment

Both authors are grateful to the referee whose comments improved the presentation of our results.