TL;DR: In this article, it was shown that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or to a proper algebraic subgroup of G.
Abstract: We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromov's proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
TL;DR: The instability of sociometric group assignments completed with the standard score and probability methods indicates that researchers should be cautious about the use of classifications based on only one data collection and that the selection of children for both clinical intervention and further nomothetic research may require alternative assessment procedures.
Abstract: The stability of standard score and probability method sociometric group assignments was examined over a 2-year period with an initial group of 334 preadolescents. The popular, neglected, and controversial sociometric groups evidenced low stability of group membership over intervals of approximately 1, 6, 12, 18, and 24 months; the rejected group evidenced slightly higher short-term stability. These findings of limited stability were attributed to measurement error and to the failure of both classification systems to identify groups with homogeneous social reputation profiles. Social role scores contributed to the prediction of stable group membership in the rejected and controversial classification, although these scores added little to the prediction of stable popular and neglected group membership. Stability over short intervals could be used to enhance the prediction of stability over longer periods; however, this procedure resulted in the classification of numerous false positives and false negatives. The instability of sociometric group assignments completed with the standard score and probability methods indicates that researchers should be cautious about the use of classifications based on only one data collection and that the selection of children for both clinical intervention and further nomothetic research may require alternative assessment procedures.
TL;DR: Mennecke, Hoffer, and Wynne as mentioned in this paper tested three models of change and continuity in group structure using existing longitudinal data on 20 small groups, meeting face to face or via a computer-mediated communication system for 13 weeks.
Abstract: Three models of change and continuity in group structure are tested using existing longitudinal data on 20 small groups. Groups met face to face or via a computer-mediated communication system for 13 weeks. Computer-mediated groups fit the robust equilibrium pattern best, with initial fluctuations in the influence hierarchy followed by a more stable structure that persisted despite changes in operating conditions. Face-to-face groups fit a bistable punctuated equilibrium pattern best, retaining their initial influence structure until an environmental cue triggered a shift. Contrary to die predictions of this model for radical change, adjustments were modest. Poor performance on tasks failed to trigger changes predicted by the adaptive response model, probably because outcomes were not very important to group members. When a number of persons have come together to form a group, their behavior never holds to its first pattern. Homans, 1950, p. 109 Lasting patterns can appear as early as the first few seconds of a group's life. Gersick, 1988, p. 33 Traditional models of group development—the patterning of change and continuity in group structure and behavior over time—propose that groups follow a fixed sequence of stages (Bales, 1970; Bennis & Shepard, 1956; Tuckman, 1965; Tuckman & Jensen, 1977; see Mennecke, Hoffer, & Wynne, 1992, for a review). These models typically presume a stable group composition and context and ignore external causes of change. Changes in group composition or other interventions are treated as externalities that disrupt the ideal path of development (Hill & Gruner, 1973). Yet most naturally occurring groups with an extended history experience temporary or permanent changes of membership and cope (or fail to cope) with other changes
TL;DR: This paper developed a positive analysis of stable group formation, highlighting the role of conflict management within groups, based on a simple economic model that features a "winner-take-all" contest for control of some resource.
Abstract: This paper develops a positive analysis of stable group formation, highlighting the role of conflict management within groups. The analysis builds on a simple economic model that features a “winner-take-all” contest for control of some resource. When a group forms, members pool their efforts in that contest and, if successful, apply the resource to a joint production process. While reducing the severity of conflict over the contestable resource relative to the case of individual conflict, the formation of groups adds another layer of conflict, that is, one between the members of the winning group over the distribution of their joint product. The effectiveness of conflict management in enabling groups to resolve this second layer of conflict in more “peaceful” ways has some important implications for the equilibrium structure of groups as well as for the allocation of resources.