Abstract: We will introduce new group invariants inspired by algebraic number theory and investigate how effective they are in distinguishing finite groups. Only basic definitions concerning groups such as cosets (and a little bit of group actions) will be assumed. Algebraic number theory itself will be conspiciously absent. The point of the talk will be whether y'all can to tell me if you have seen this sort of thing before, and/or suggest methods from graph theory to advance the problem.