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 alg...

In the recent progress [BE1], [Me] and [Z2], the wellknown JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd or...

We shall say that an automorphism a is nilpotent or acts nilpotently on a group G if in the holomorph H= [G](a) of G with a, a is a bounded left Engel element, that is, [H, ¿a] = l for some natural number ¿. Here [H, ka] means [H, (k — l)a] with [H, Oa] denoting H. Let G' denote the commutator subgroup [G, G], and let $(G) denote the Frattini subgroup of G. If a is an automorphism of a nilpoten...

Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentially nilpotent over F with a special emphasis on the case that F = Zp is a finite field. A necessary condition on F is observed for...

Let N be a nilpotent group normal in a group G. Suppose that G acts transitively upon the points of a finite non-Desarguesian projective plane P. We prove that, if P has square order, then N must act semi-regularly on P. In addition we prove that if a finite non-Desarguesian projective plane P admits more than one nilpotent group which is regular on the points of P then P has non-square order a...

Let O be a nilpotent orbit in the Lie algebra n( ) and let V be an orbital variety contained in O. Let P be the largest parabolic subgroup of SL(n, ) stabilizing V. We describe nilpotent orbits such that all the orbital varieties in them have a dense P orbit and show that for n big enough the majority of nilpotent orbits do not fulfill this. Résumé Soit O une orbite nilpotente dans l’algèbre de...

We determine the admissible nilpotent coadjoint orbits of real and p-adic split exceptional groups of types G2, F4, E6 and E7. We find that all Lusztig-Spaltenstein special orbits are admissible. Moreover, there exist nonspecial admissible orbits, corresponding to “completely odd” orbits in Lusztig’s special pieces. In addition, we determine the number of, and representatives for, the non-even ...

An automorphism of a group G is called regular if it moves every element of G except the identity. BURNSIDE proved that a finite group G has a regular automorphism of order two if and only if G is an abelian group of odd order, and then the only such automorphism maps every element onto its inverse ([21, p. 230). More recently several authors considered the question: what groups can admit regul...

The genus of a finitely generated nilpotent group G is defined as the set of isomorphism classes of finitely generated nilpotent groups K such that the p-localizations Kp, Gp are isomorphic for all primes p [19]. This notion turns out to be particularly relevant in the study of non-cancellation phenomena in group theory and homotopy theory. In the above definition, the restriction of finite gen...