Eggert’s Conjecture on the Dimensions of Nilpotent Algebras

In 1971, Eggert [2] conjectured that for a finite commutative nilpotent algebra A over a field K of prime characteristic p > 0, dimA ≥ p dimA(p), where A(p) is the subalgebra of A generated by all the elements xp, x ∈ A and dimA, dimA(p) denote the dimensions of A and A(p) as vector spaces over K. In [3], Stack conjectures that dimA ≥ p dimA(p) is true for every finite dimensional nilpotent algebra A over K . We point out that some particular cases of Eggert’s conjecture have been proved in [1, 2, 3, 4]. Here we prove the conjecture for finite dimensional commutative nilpotent algebras. This combined with the results of [2] completely describe the group of units of A and the problem set in [1]: “When a finite abelian group is isomorphic to the group of units of some finite commutative nilpotent algebras?” is solved. Recall that the group of units of A is the set A with the following operation: x · y = x+ y + xy, ∀x, y ∈ A.