Non-isomorphic nilpotent associative algebras may have isomorphic algebra groups

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Statement

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.

Proof

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

Proof for the prime two

See direct product of Z4 and Z2 is an algebra group for two non-isomorphic nilpotent associative algebras.