The n-Category Café

I have seen groupoids introduced early on, perhaps not right at the start. (Where … where else Bangor!)The theorems that work well from a groupoid point of view are the Nielsen Schreier theorem and the Schreier index theorem. These can be found in work by Phil Higgins for instance probably in his reprinted categories and groupoids book (TAC reprints).

I have used groupoids as a tool in courses on group presentation theory. They work well there. Also see Ronnie Brown’s Topology and Groupoids especially chapter 10.

Not group theory, but I used, with some success, Brown’s Topology and Groupoids to teach my two high school kids about basic homotopy theory back when I was still tutoring bright high school kids.

Why not start from quasigroups/latin square instead ?

You begin with some tiny combinatorics considerations, (quasigroups and loops as latin square) and then introduce further axioms giving your students appreciation for the regularizing properties and advantages of associativity that they take for granted with relative integers and fractions.

A more technical way in this trend is the book “Postmodern Algebra” coauthored by J.D.H. Smith.