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In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain condition (that is, all ascending/descending chains in this family are finite). What I'm interested in is weaker 'virtual' chain conditions: a virtual DCC would be that given a sequence $G_1 > G_2 > \dots$ of subgroups (of some special kind) such that $G_{i+1}$ has infinite index in $G_i$ for all $i$, then the sequence must terminate. One can define virtual ACCs similarly.

Does anyone know of work that has been done on conditions of this kind, either showing that a family of subgroups satisfies the conditions or deriving consequences from them? References for analogous conditions in other algebraic contexts would also be interesting.

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The virtual DCC doesn't seem so different from the notion of Krull dimension $1$ that I explained in answer to this Different definitions of the dimension of an algebra question.

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