I think you mean the multiplicative group of units of Z/nZ, because additively Z/nZ has much more structure than a group. Still, for my example and in the artic

I think you mean the multiplicative group of units of Z/nZ, because additively Z/nZ has much more structure than a group. Still, for my example and in the article, one should be motivated to study groups because they show up in places that you don't expect to find algebraic structure. Z/nZ and its variants are the obvious places to look. Elliptic curves are surprising. Also, I wrote a series on elliptic curves which doesn't need any graduate level mathematics to appreciate [1]. Of course if you want to make breakthroughs on elliptic curves (or any serious mathematical topic) you need graduate study. [1]: http://jeremykun.com/2014/02/08/introducing-elliptic-curves/ http://jeremykun.com/2014/02/08/introducing-elliptic-curves/