Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 67).

In this thesis we investigate the structure of locally nilpotent 5-Engel p-groups. We show that for p > 7, locally nilpotent 5-Engel p-groups have class at most 10. This is a global theorem, where the result is not dependent on the number of generators of the group. The proof uses new and established Lie methods and a custom C++ implementation of an algorithm that constructs minimal generating sets and structure constants of multi- graded Lie algebras in a variety defined by three multilinear relations, which hold in the Lie rings associated with 5-Engel p-groups. We obtain our results by calculating in the set Q(p) = {~ I x E Z, yE Z+, Y # 0 modulo any p f/. p} (where p is a set of excluded primes and x, y are arbitrarily large integers), as well as the fields Zp, p prime. We introduce several reduction theorems, making the result possible. We also present results about the normal closure of elements in these groups. We use a Higman reduction theorem and the same custom C++ program to show that locally nilpotent 5-Engel p-groups, p 2: 5, are Fitting, with Fitting degree at most 4 if p > 7, at most 5 if p = 7 and at most 6 if p = 5. These results are best possible.

In this thesis we find deep results on the structure of normal right n-Engel subgroups that are contained in some term of the upper central series of a group. We start with some known results, and one new result, on the structure of locally nilpotent n-Engel groups. These are closely related to the solution of the restricted Burnside problem. We also give specific details of the structure of 2-Engel and 3-Engel groups in the context of these results. The main idea of this thesis is to generalise these results to apply to normal upper central right n-Engel subgroups. We also consider the special case of locally finite p-groups and again generalise some deep results on the structure of n-Engel such groups to apply to right n-Engel subgroups. For each of the theorems on right n-Engel subgroups, complete details are given for the case n = 2. Right 3-Engel subgroups have a more complicated structure. For these we prove a Fitting result, for which we exclude the prime 3, and using this we also find a sharp bound on the upper central degree in the torsion-free case. In fact we only need to exclude the primes 2, 3 and 5 for this result. This gives some further information on the structure of right 3-Engel subgroups in the context of the main theorems.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (doctoral)--Universität Bonn, 1988. / Includes bibliographical references.

Thesis (Ph. D.)--Tohoku University, 2004. / "June 2004." Includes bibliographical references (p. 69-72).

In this thesis, we introduce the theory of locally nilpotent derivations and use it to compute certain ring invariants. We prove some results about quasi-extensions of derivations and use them to show that certain rings are non-rigid. Our main result states that if k is a field of characteristic zero, C is an affine k-domain and B = C[T,Y] / < T^nY - f(T) >, where n >= 2 and f(T) \in C[T] is such that delta^2(f(0)) != 0 for all nonzero locally nilpotent derivations delta of C, then ML(B) != k. This shows in particular that the ring B is not a polynomial ring over k.

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency, as decomposed into the lower central series and eventual zero subspace. The map sending the Lie algebra L to a derivation of L is called the adjoint representation, where a given Lie algebra is nilpotent if and only if the adjoint is nilpotent. Our goal is to prove Engel's Theorem, which states that if all elements of L are ad-nilpotent, then L is nilpotent.

This thesis is concerned with the classification of 7-dimensional nilpotent Lie algebras. Skjelbred and Sund have published in 1977 their method of constructing all nilpotent Lie algebras of dimension <i>n</i> given those algebras of dimension < <i>n</i>, and their automorphism groups. By using this method, we construct all nonisomorphic 7-dimensional nilpotent Lie algebras in the following two cases: (1) over an algebraically closed field of arbitrary characteristic except 2; (2) over the real field <strong>R</strong>. We have compared our lists with three of the most recent lists (those of Seeley, Ancochea-Goze, and Romdhani). While our list in case (1) over <strong>C</strong> differs greatly from that of Ancochea-Goze, which contains too many errors to be usable, it agrees with that of Seeley apart from a few corrections that should be made in his list, Our list in case (2) over <strong>R</strong> contains all the algebras on Romdhani's list, which omits many algebras.

This thesis is concerned with the classification of 7-dimensional nilpotent Lie algebras. Skjelbred and Sund have published in 1977 their method of constructing all nilpotent Lie algebras of dimension <i>n</i> given those algebras of dimension < <i>n</i>, and their automorphism groups. By using this method, we construct all nonisomorphic 7-dimensional nilpotent Lie algebras in the following two cases: (1) over an algebraically closed field of arbitrary characteristic except 2; (2) over the real field <strong>R</strong>. We have compared our lists with three of the most recent lists (those of Seeley, Ancochea-Goze, and Romdhani). While our list in case (1) over <strong>C</strong> differs greatly from that of Ancochea-Goze, which contains too many errors to be usable, it agrees with that of Seeley apart from a few corrections that should be made in his list, Our list in case (2) over <strong>R</strong> contains all the algebras on Romdhani's list, which omits many algebras.