Chain Conditions for Subgroups of Infinite Order or Index

A group G is said to satisfy max-∞ if each nonempty set of infinite subgroups of G has a maximal element. A group G is said to satisfy min-∞ if each nonempty set of subgroups of G with infinite index has at least one minimal element. Groups with max-∞ or min-∞ are the subject of this paper. Abelian, nilpotent, and solvable groups with max-∞ or min-∞ are examined in detail and structure theorems are given in each case. We then characterize the groups with max-∞ or min-∞ 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.