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This paper deals with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions. If P is a theoretical property of groups and subgroups, we show that a locally graded group P satisfies the minimal conditions for subgroups not having P if and only if either G is a Cernikov group or every subgroup of G satisfies P, for certain values of P concerning normality, nilpotency and related ideas.
In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration. We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups.
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