The purpose of this paper is to describe the process by which a pair of undergraduate students, participating in a teaching experiment, reinvented (with guidance) the concepts of group and isomorphism beginning with an exploration of the symmetries of an equilateral triangle. The intent of this description is to highlight some important insights provided by an analysis of the students' mathematical activity. First, the analysis resulted in the identification of a number of informal student strategies that anticipated the formal concepts. Second, the analysis provided insight into how these strategies could be evoked. Third, the analysis provided insight into how these strategies could be leveraged to support the development of the formal concepts. (Contains 16 figures.)