# Non-isomorphic Heisenberg groups over rings

Suppose $R_1,R_2$ are finite unital commutative rings. Consider Heisenberg groups $H_3(R_1)$ and $H_3(R_2)$ (upper unitriangular marticies $3 \times 3$).

Proposition. If $R_1 \not\cong R_2$ (as rings) then $H_3(R_1) \not\cong H_3(R_2)$ (as multiplicative groups).

Is that true? It seem that if $R_1$ and $R_2$ have not isomorphic additive groups then $H_3(R_1) \not\cong H_3(R_2)$, since they have not isomorphic centers. But what about general case, or is that too broad?

However, Proposition 1.9 in the same paper asserts that this does not extend to the non-commutative (associative unital case). The counterexamples have the form $R_1=K\times K$, $R_2=K\times K^{\mathrm{op}}$, where $K$ is indecomposable and not isomorphic to $K^{\mathrm{op}}$.