# Non-isomorphic nilpotent associative algebras may have isomorphic algebra groups

Let $q$ be any prime power. It is possible to construct associative algebras $N_1$ and $N_2$ over $\mathbb{F}_q$, both of which are nilpotent (i.e., all products of a certain length or more are zero), such that $N_1$ and $N_2$ are not isomorphic as $\mathbb{F}_q$-algebras, but the algebra group corresponding to $N_1$ is isomorphic to the algebra group corresponding to $N_2$.