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**Miroslav Korbelar ^{*}**

Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovska, Czech Republic

- Corresponding Author:
- Miroslav Korbelar

Faculty of Mathematics and Physics

Department of Algebra, Charles University

Sokolovska 83, 186 75 Prague 8, Czech Republic

**Tel:**+420 224 491 111

**E-mail:**[email protected]

**Received date:** July 24, 2015 **Accepted date:** August 03, 2015 **Published date:** August 31, 2015

**Citation:** Korbelar M (2015) Eggert’s Conjecture for 2-Generated Nilpotent Algebras. J Generalized Lie Theory Appl S1:001. doi:10.4172/1736-4337.S1-001

**Copyright:** © 2015 Korbelar M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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 ap , a ∈ A. We show that the conjecture holds if A(p) is at most 2-generated.

Nilpotent algebra; Eggert's conjecture; Commutative nilpotent ring; Polynomial bases

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

p· dim A^{(p)} ≤ dim A.

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:

(i) p = 2 and (A^{(p)})^{4} = 0.

(ii) A^{(p)} is 2-generated.

(iii) (A^{(p)})^{3} = 0.

(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], resp.) the algebra (vector space, resp.) generated by X.

An algebra A is called nilpotent if Am=0 for some m ∈ N.

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.

We will use the well-known concept of monomial ordering and standard bases.

For put

Denote the multiplicative monoid with the lexicographical ordering ≤ such that

and

x^{(i,j)} ≤ 0

for every

For

Finally, f will be called normal iff λ_{α0} = 1, where m(f) = x^{α 0}, and m(f) < π x^{α} implies λ_{α} = 0 for every

This function m: F [x, y] → [X]_{0} has common properties of a valuation:

(i) m(fg) = m(f) m(g).

(ii) m(f + g) ≥ min{m(f); m(g) g. Moreover, m(f + g) = m(f) if m(f) < m(g).

(iii) m (f(x^{p}, y^{p})) = m(f)^{p}.

for every f, g∈ F [x, y].

Finally. a set will be called upper (lower, resp.) if implies

**Definition 2.1.** Let A be a nilpotent F -algebra generated by {a_{1}, a_{2}}. Put

and

**Proposition 2.2.** *Let A be a nilpotent F -algebra generated by {a _{1}, a_{2}}. Then*

(i)

(ii) *is a lower set and *1∈

(iii)The set * is a basis of A. In particular, is finite.*

(iv)

**Definition 2.3.** Let A be a nilpotent F -algebra generated by {a_{1}, a_{2}}. Denote

and

for

**Lemma 2.4.*** Let A be a nilpotent F -algebra generated by {a _{1}, a_{2}}. Then:*

Let I ⊆ Rx + Ry be an ideal in R such that A = Rx + Ry/I is a nonzero nilpotent F -algebra.

We have A = and A^{(p)} =

By definition of there are such that mare normal.

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 yk, 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 number (see 3.4 and the definition of B_{A}^{(p)}(a_{1}^{p},a_{2}^{p})

**Lemma 3.1. **

**Definition 3.2. **Denote

wA = max B_{A}A(x + I, y + I).

For denote

the least integer such that Put

Following lemma is obtained using induction.

**Lemma 3.3.** *Let 1≤ i ≤ n _{0} + 1 and 0 ≠ f ∈ I be such that m(f) x^{i}. Then*

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)}.

**Proposition 3.4.**

Now, only exploring carefully the previous cases for i and li we get the following interesting claim. It says that the inequality holds for almost every i.

**Theorem 3.5. ***One of the following cases takes place:*

And our main result is just an easy corollary of this and 1.1.

**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.

- Eggert N (1971)Quasi regular groups of finite commutative nilpotent algebras. Pacific J Math 36: 631-634.
- Bautista R (1976) Units of finite algebras. An. Inst. Mat. Univ. Nac.Autonoma. Mexico 16: 1-78.
- Stack C (1996) Dimensions of nilpotent algebras over fields of prime characteristic. Pacific J Math176: 263-266.
- Stack C (1998)Some results on the structure of finite nilpotent algebras over field of prime characteristic.J. Combin. Math.CombinComput 28: 327-335.
- Amberg B and Kazarin LS (2001) Commutative nilpotent p-algebras with small dimension.Quaderni di Mat. (Napoli) 8: 1-20.
- McLean KR (2004)Eggert's conjecture on nilpotent algebras.Comm Algebra 32: 997-1006.
- McLean KR (2006) Graded nilpotent algebras and Eggert's conjecture.Comm Algebra 34: 4427-4439.
- Gorlov VO (1995) Finite nilpotent algebras with metacyclicadjoint group.Ukrain Math Z 47: 1426-1431.
- Hammoudi L (2002)Eggert's conjecture on the dimensions of nilpotent algebras. Pacific J Math 202: 93-97.
- AmbergB and Kazarin LS (2005) Nilpotent p-algebras and factorized p-groups. Proceedings of Groups St. Andrews 1: 130-147.
- Korbelar M (2010) 2-ge nerated nilpotent algebras and Eggert's conjecture. Journal of Algebra 324:1558–1576.

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