Chain conditions for subgroups of infinite order or index

A group G is said to satisfy max-infinity if each nonempty set of infinite subgroups of G has a maximal element. A group G is said to satisfy min-infinity if each nonempty set of subgroups of G with infinite index has at least one minimal element. Groups with max-infinity or min-infinity are the subject of this paper. Abelian, nilpotent, and solvable groups with max-infinity or min-infinity are examined in detail and structure theorems are given in each case. We then characterize the groups with max-infinity or min-infinity in the smallest class of groups containing all linear groups which is locally closed and closed with respect to the formation of ascending series with factors in the class. (C) 2002 Elsevier science (USA).