Nonabelian infinite nilpotent groups

Question

Can you give examples of nonabelian infinite nilpotent groups?

Here's what I got so far:

1. The Heisenberg group.
2. The free nilpotent group of class $s$ (thanks Arturo for your comment here).
3. 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.

Answer 1

For example, these are constructed in Robinson's A Course in the Theory of Groups, 2nd Edition:

1. Let $R$ be a ring (not necessarily commutative, not necessarily with identity); if $n\gt 0$, then we let $R^{(n)}$ be the set of all sums of products of $n$ elements of $R$; then $R^{(n)}$ is a subring of $R$, and $R^{(n)}=0$ if and only if every product of $n$ elements of $R$ is equal to $0$. If $R^{(n)}=0$ for some $n\gt 0$, we say that $R$ is a nilpotent ring.

   Now let $S$ be a ring with identity, and assume that $R$ is a subring of $S$ that is a nilpotent ring (in particular, $R$ does not contain the identity). Let $U$ be the set of all elements of the form $1+x$ with $x\in R$. Then $U$ is a group under the operation given by multiplication in $S$: that is, given $1+x$ and $1+y$ in $U$, then $$(1+x)(1+y) = 1 + (x + y + xy) \in U$$ and $$(1+x)^{-1} = 1 + (-x + x^2 - x^3 + x^4 -\cdots + (-1)^{n-1}x^{n-1})\in U,$$ where $x^n = 0$.

   Then $U$ is a nilpotent group, and if we let $U_i = R^{(i)}$, then $$1 = U_n \leq U_{n-1}\leq\cdots\leq U_1 = U$$ is a central series for $U$.

   For example, let $S$ be the ring of $n\times n$ matrices over an infinite commutative ring with identity $T$, and let $R$ be the set of upper zero triangular matrices; then $R$ is nilpotent, and the corresponding $U$ is the group of upper unitriangular matrices over $T$, which is nilpotent of class exactly $n-1$. (This generalizes the Heisenberg group).

2. Let $A$ be a nontrivial abelian group, let $D=A\times A$, and let $\delta\colon D\to D$ be given by $\delta(a,b) = (a,ab)$. Let $G=D\rtimes\langle \delta\rangle$. Then $G$ is nilpotent of class $2$. If $A$ is infinite, so is $G$.