[For the entire collection see Zbl 0542.00008.]
In the theory of infinite groups, the notion of ”largeness” has not always been defined concisely. A previous work of the second author added precision to this notion [Word Problems II; Stud. Logic Found. Math. 95, 299-335 (1980; Zbl 0438.20023)]. The purpose of this article is to survey most of the known results as well as to develop some new ones on ’largeness’ subject to a modification in the earlier definition. A large property is to be a group-theoretic property \({\mathcal P}\) that satisfies the following conditions: (i) If a group G has \({\mathcal P}\) and the group K maps onto G, then K has \({\mathcal P}\). (ii) If H is a subgroup of finite index in a group G, then G has \({\mathcal P}\) if and only if H has \({\mathcal P}\). (iii) If N is a finite normal subgroup of a group G with \({\mathcal P}\), then G/N has \({\mathcal P}\). The scope of the work prevents elaboration on the details in this review. It should suffice to comment that the article is relatively self-contained, carefully presented, and contains many excellent examples. For completeness, open problems are identified. The concept that is developed here is refreshing.