Journal of Generalized Lie Theory and Applications Eggert's Conjecture for 2-Generated Nilpotent Algebras

Let A be a commutative nilpotent finitely-dimensional algebra over a field F of characteristic p > 0. A conjecture of Eggert says that p · dim A (p) dim A, where A (p) is the subalgebra of A generated by elements a p , a ∈ A . We show that the conjecture holds if A (p) is at most 2-generated.


Introduction
Let F be a field of characteristic p>0 and A a commutative (associative) nilpotent finite-dimensional algebra over F . Let A (p) be the subalgebra generated by the set {a p | a ∈ A}. N. Eggert [1] conjectured that This conjecture gives an answer to the problem, when a finite abelian group is isomorphic to the adjoint group of some finite commutative nilpotent F-algebra. Recall that the adjoint group of A is the set A with the operation x○ y=x + y + x y for every x, y ∈ A.
Validity of this hypothesis would also have influence on an estimation of a (Prüfer) rank of a product of two (abelian) p-groups.
N. Eggert proved his conjecture only when dim A (p) ≤ 2. Five years later, R. Bautista [2] proved it when dim A (p) = 3. C. Stack confirmed this results in Stack et al. [3,4], but provided shorter proofs. Finally, Amberg and Kazarin [5] proved the conjecture for the case dim A (p) ≤ 4.
Another type of results presented by McLean [6,7]. He showed that this conjecture is true if the algebra A is either radical of a group algebra of a finite abelian group or A is graded and at least one of the following conditions is fulfilled: (iv) n < 3p and 3 ≤ s -1 ≤ p, where n is the number of generators of A (p) and s is the index of nilpotence of A (p) .
We also should mention the result of Gorlov [8]. He proved the conjecture for nilpotent algebras A with a metacyclic adjoint group.
One paper concerning Eggert's conjecture appeared in 2002 and the author L. Hammoudi [9] claimed he proved it. But, as Amberg [10] and McLean [7] have shown, his proof was incorrect.
In this short note we sketch out the main steps of the proof that Eggert's conjecture is true if the subalgebra A (p) has at most two generators. For the details, the reader is referred to Korbelar [11].
Since we will deal with nilpotency and commutativity only, we point out that the word 'algebra' will mean a commutative one and not necessary possesing a unit.
For an algebra A and a subset X ⊆ A we denote X ([X], resp.) the algebra (vector space, resp.) generated by X.
Through this paper let always F be a field of characteristic p > 0 and R = F [x, y] be the ring of polynomials over the variables x, y and the field F.
We start with the remark, that the number of any minimal generating set of a finite generated nilpotent F -algebra A is equal to dim A/A 2 . This implies the following:

Lemma 1.1. Suppose that Eggert's conjecture holds for every nilpotent 2-generated F -algebra. Then it also holds for every nilpotent F -algebra A such that A(p) is a 2-generated F -algebra.
In the rest we deal with 2-generated nilpotent algebras.

Bases of Nilpotent Algebras
We will use the well-known concept of monomial ordering and standard bases.
(ii) A (a 1 , a 2 ) is a lower set and 1∈  A (a 1 , a 2 ).

Eggert's Conjecture for 2-generated Algebras
Let I ⊆ Rx + Ry be an ideal in R such that A = Rx + Ry/I is a nonzero nilpotent F -algebra. The main idea of the proof lies in the fact that taking a normal polynomial from I, dividing it by x and then multiplying by some suitable y k , we get again a member of I (3.3). Then, using binomial formula in a suitable way, we obtain a polynomial that will estimate the  Following lemma is obtained using induction.
The proof of the next proposition uses only the binomial formula. It finds the particular polynomial the we need to make an estimation of the numbers D i and thus of the dimension of A (p) .   Now, only exploring carefully the previous cases for i and l i we get the following interesting claim. It says that the inequality " Theorem 3.6. Let A be a nilpotent F -algebra, char F=p>0, such that A (p) is 2-generated. Then p·dim A (p) dim A.