  We give a simple matrix-based proof of congruence equations modulo a prime
$p$ involving sums of binomial coefficients appearing in Pascal's triangle.
These equations can be used to construct some groups of exponent $p^n$. These
groups, as well as others of exponent $p^{n+1}$, explain why $p=2$ is not
really an exceptional prime in relation to the Heisenberg group over the field
with $p$ elements.
