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 T is the class of all finite groups, which is the union of an ascending sequence of quasi-varieties, then L(T) not equal N(T). Examples of finitely generated groups in L(T)N(T) are also constructed.
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