## February 17, 2009

### Can -oids save Group Theory 101?

#### Posted by David Corfield

Can you help Lieven out with some questions?

Have you seen a first-year group theory course starting off with groupoids? Do you know an elegant way to prove a classical group-result using groupoids?

Posted at February 17, 2009 5:04 PM UTC

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### Re: Can -oids save group-theory 101?

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.

Posted by: Tim Porter on February 17, 2009 5:38 PM | Permalink | Reply to this

### Re: Can -oids save group-theory 101?

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.

Posted by: Mikael Vejdemo Johansson on February 17, 2009 6:50 PM | Permalink | Reply to this

### Re: Can -oids save Group Theory 101?

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.

Posted by: Octave Schwimmer on February 23, 2012 11:57 AM | Permalink | Reply to this

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