Yep! And I really wish this concept (of bringing together all of those properties, and exploring some of the resulting structures) was taught (specifically including the introduction of the term "group") in high school algebra. I think that would reduce a lot of the friction between learning secondary and post-secondary mathematics.

It _is_ taught to 16 year olds in Australia, if they elect to take math C (math a/B is compulsory where math A is remedial). Everyone hated it and wondered what the point was (including me - it was only when I did a course on abstract algebra in university and learned about quotient groups that I was converted)

Also covered in math C is the computational parts of linear algebra (e.g. Gaussian elimination, Cramer's rule).

I took the International Baccalaureate (IB) programme when I was in my final two years of High School (September 1997-June 1999) and our Maths Higher unit included ”Group Theory”.