Can you give examples of nonabelian infinite nilpotent groups?
Here's what I got so far:
- The Heisenberg group.
- The free nilpotent group of class $s$ (thanks Arturo for your comment here).
- The group of (some) symmetries of polynomials of degree up to by $s$ generated by the symmetry which adds a constant polynomial (for each constant polynomial) and translation of the argument of the polynomial by a scalar (for each scalar).
(got the last 2 examples from http://terrytao.wordpress.com/2009/12/21/the-free-nilpotent-group/).
I'm looking for more examples of such groups. Infinite nonabelian groups which are not nilpotent, but have a nilpotent subgroup of finite index are also of interest to me.