A group G is said to satisfy max-infinity if and only if each nonempty set of infinite subgroups of G has a maximal element or equivalently, if and only if there does not exist an infinite properly ascending chain G1 G2 ··· of infinite subgroups. Groups satisfying max-infinitys or max-infinityn can be defined similarly by imposing the condition on nonempty sets of infinite subnormal or infinite normal subgroups respectively.;A group G is said to satisfy min-infinity if and only if each nonempty set of subgroups of G with infinite index has at least one minimal element or equivalently, if and only if there does not exist an infinite properly descending chain G 1 > G2 > ··· of subgroups of infinite index in G. Again, we can define groups satisfying min-infinitys or min-infinity n analogously.;Groups with max-infinityi or min-infinityi for i=empty,s,n are the subject of this dissertation. 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. In addition, we discuss characterizations of solvable groups and subsolvable groups with max-infinitys or min-infinitys. Investigations of locally nilpotent and metanilpotent groups with max-infinityn or min-infinity n are included as final topics.
展开▼
