Locally Finite-Indicable Groups

Abstract

A group is locally ℜ-indicable if every finitely generated subgroup has a nontrivial homomorphism onto a nontrivial ℜ-group. If ℜ is a quasi-variety, then the class L(ℜ) of locally ℜ-indicable groups coincides with the class N(ℜ) of groups which have normal systems with factors in ℜ. It is not known if ℜ must be a quasi-variety in order for the equality L(ℜ) = N(ℜ) to hold. We show here that if ℑ is the class of all finite groups, which is the union of an ascending sequence of quasi-varieties, then L(ℑ) ≠ N(ℑ). Examples of finitely generated groups in L(ℑ)\ N(ℑ) are also constructed.

Key Words:

• Infinite alternating group
• Locally indicable groups
• Normal systems
• Quasi-varieties

Mathematics Subject Classification:

• 20E25
• 20E10

Notes

Communicated by S. K. Sehgal