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

From Groupprops

## Statement

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

## Proof

Currently, we only have the proof for the prime two, but a similar proof technique should work in other cases.