Prove that a group law of the Heisenberg group is continuous.

Question

The group law on $\mathbb{H}^n$ -Heisenberg group- is given as follows: $(s,x,y)·(s′,x′,y′)=(s+s′+ ω(x,y;x′,y′),x+x′,y+y′)$,

how can I prove that this group law is continuous?

Many thanks.

Answer 1

There's always, "Let $\epsilon > 0$ ...", but you might find it easier to observe that the Heisenberg group can be realized as the upper triangular 3x3 matrices with 1s on the diagonal, and the group operation can be realized by matrix multiplication. Then continuity of the group operation follows from continuity of matrix multiplication.