On the adjoint group of a finite nilpotentp-algebra (original) (raw)

3 ADJOINT GROUPS OF p-NIL RINGS AND p-GROUP AUTOMORPHISMS

2016

We introduce a class of rings, namely the class of left or right p-nil rings, for which the adjoint groups behave regularly. Every p-ring is close to being left or right p-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of p-groups. These facts and some of their applications are investigated in this paper.

On the Dimension of Nilpotent Algebras

2001

The Eggert conjecture claims that a finite commutative algebra R over a field of prime characteristic p has the property dim R ≥ p dim R (1) , where R (1) is the subspace of R spanned by the pth powers of elements of R . We obtain results related to this conjecture and results on nilpotent algebras of rather high nilpotency class.

Adjoint groups of ppp-nil rings and ppp-group automorphisms

Bulletin of the Belgian Mathematical Society, Simon Stevin

We introduce a class of rings, namely the class of left or right ppp-nil rings, for which the adjoint groups behave regularly. Every ppp-ring is close to being left or right ppp-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of ppp-groups. These facts and some of their applications are investigated in this paper.

Nilpotency: A Characterization Of The Finite p-Groups

Journal of Mathematics , 2017

Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]

On the subring structure of finite nilpotent rings

Pacific Journal of Mathematics, 1969

This paper studies the nilpotent ring analogues of several well-known results on finite p-groups. We first prove an analogue for finite nilpotent p-rings [a ring is called a p-ring if its additive group is a p-group] of the Burnside Basis Theorem, and use this to obtain some information on the automorphism groups of these rings. Next we obtain Anzahl results, showing that the number of subrings, right ideals, and two-sided ideals of a given order in a finite nilpotent p-ring is congruent to 1 mod p. Finally, we characterize the class of nilpotent p-rings which have a unique subring of a given order. The analogy between nilpotent groups and nilpotent rings which motivates the results of this paper is the replacement of group commutation by ring product. A nilpotent ring, of course, is itself a group under the circle composition x o y = x + y + xy but the structure of this group implies little about the invariants to be studied here, as shown by the examples in the last section of the paper.

On Nilpotent Groups of Algebra Automorphisms

Nagoya Mathematical Journal

The main purpose of this paper is to derive conclusions about the structure of a nilpotent group of algebra automorphisms and, in the case of a Lie algebra, about the influence of this nilpotence on the structure of the algebra. A motivation for this study is a well known theorem due to Kolchin: A unipotent linear group can be triangularized and is thus nilpotent. The converse is manifestly false, but we have (as an immediate consequence of Theorem 2.7):

Some necessary and sufficient conditions for p-nilpotence of finite groups

Bulletin of the Australian Mathematical Society, 2003

The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.

On a Class of Generalized Nilpotent Groups

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

The algebraic classification of nilpotent algebras

Journal of Algebra and Its Applications, 2021

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.