The group model is the application's internal representation of the group, in the same way as the "user model" is an application's internal model of the user.
By letting the model change, and choosing functionality and behaviour appropriately, the application can better match the user's needs.
However if the model is constructed incorrectly or too obviously, the application starts obviously mediating transactions and gets in the way.
Despite the research of Reeves and Nass in The Media Equation, just because people perceive the computer as an actor, if the computer is too obviously "present" it's annoying -- consider Microsoft Bob, or Clippy in Microsoft Help for example (MS funded some of Reeves and Nass' research. Good on them for taking an interest, but I think they took it too literally).