I always had a strong belief that principles of Group Theory must correlate into writing type systems.

Just looked at the ToC of this book and there is an entire chapter on Groups in that book ! Not only Groups but also goes into Rings as well...

Must read this book.

In a Group, every element has an inverse element. If you drop this property, you get a Monoid.[0] Functions under function composition form a Monoid (or Monoidal Category). Brian Beckman did a nice intro video to this at MS Channel 9 a few years back.[1]

[0] https://en.wikipedia.org/wiki/Monoid

[1] "Brian Beckman: Don't fear the Monad" https://www.youtube.com/watch?v=ZhuHCtR3xq8

Groupoids, a generalization of groups, factor heavily into type theory!

I think what you want is Category Theory.

Question -- clearly, when you look at Haskell, you see abstract algebra. Were Stepanov's ideas big influencers of Haskell, or vice versa, or neither?

Stepanov and Musser were working on generic programming in the 80s[0][1], which predates Haskell. I suspect parametric polymorphism was common in the FP world by that time, but I don't know whether it was used as an organising principle to the extent that Stepanov promotes.

[0] https://en.wikipedia.org/wiki/Alexander_Stepanov

[1] https://en.wikipedia.org/wiki/David_Musser