Sums of binomial coefficients modulo p and groups of exponent pⁿ

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 pⁿ. These groups, as well as others of exponent pⁿ⁺¹, explain why $p=2$ is not really an exceptional prime in relation to the Heisenberg group over the field with p elements.