General Impressions:
Group Cognition and Distributed Cognition examine the same 'problem' domain- they just take different angles on it.
Distributed Cognition seems that it's been around longer with more support- Group Cognition takes a different angle- it was introduced mainly by Gerry Stahl (see below) and he is somewhat 'in the process' of establishing it.
This subject relates closely with my page on Collaborative Learning.
Also, my notes from the HCI Exam on Distributed Cognition are highly relevant, specifically:
Carroll Ch. 8 (Distributed Cognition);
Carroll Ch. 12 (CSCW); and
Carroll Ch. 13 (Ethnography / Ethnomethodology)
With respect to gaming / educational gaming, I think that distributed / group cognition has been somewhat left off the map. I thought the FAS: R & D Challenges article addressed the main topics related to educational gaming almost perfectly and outlined areas / questions that need research- however, they left out distributed cognition. I think distributed cognition or group cognition might be applied very well to online gaming research (MMOGs).
Richardson discusses distributed cognition while Stahl looks at group cognition. Both are similar in that they look at the way individuals make light of their situation, interpret their actions, surroundings, etc and construct ways to deal with them (which can then be validated by others in the social group). I got the impression that group cognition focuses more on the individuals in the group, while distributed cognition focuses more on the task(s) the group is performing.
Carroll, J. M. (Ed.) (2003). HCI Models, Theories and Frameworks: Toward a Multidisciplinary Science. San Francisco: Morgan Kaufmann publishers. ISBN: 1-55860-808-7.
Carroll Ch. 8 (Distributed Cognition);
Carroll Ch. 12 (CSCW); and
Carroll Ch. 13 (Ethnography / Ethnomethodology)
Richardson, S. (2003). Knowledge Management and the Design of Distributed Cognition Systems. Proceedings of the 38th Hawaii International Conference on System Sciences, 2003. DOI: 10.1109/HICSS.2005.373
Stahl, G. (2005). Group Cognition in Computer-Assisted Collaborative Learning. Journal of Computer Assisted Learning 21 (2), 79-90. DOI: 10.1111/j.1365-2729.2005.00115.x
Stahl, G. (2006). Group Cognition: Computer Support for Building Collaborative Knowledge. MIT Press, March 2006. Available Online: http://www.cis.drexel.edu/faculty/gerry/mit/stahl%20group%20cognition.pdf