Frank Levin

Lie Metabelian Group Rings

Abstract We characterize the nonabelian groups with Lie metabelian group rings. Moreover, such group rings are shown to be strongly Lie metabelian and strongly Lie nilpotent of class at most 3.

Generating groups for nilpotent varieties

Let ℜ c denote the variety of all nilpotent groups of class ≦ c, that is, ℜ c is the class of all groups satisfying the law , where we define, as usual, and, inductively, . Further, let F k (ℜ c ) denote a free group of ℜ e of rank k. In her book Hanna Neumann ([4], Problem 14) poses the following problem: Determine d(c) , the least k such that F k (ℜ c ) generates ℜ c . Further, she suggests, incorrectly, that d(c) = [c/2] + l. However, as we shall prove here, the correct answer is d(c) = c—1, for c ≦ 3. 2 More generally, we shall prove the following result.

On Lie metabelian group rings

Let G be a finite group without elements of orders two and three and R be a commutative ring with characteristic different from 2. If either the subrings A of R(G), the group ring of G over R, generated by the set {g + g−1; g ∈ G} or B generated by the set {g − g−1; g ∈ G} is Lie metabelian, then G is abelian.

Generating groups of certain soluble varieties

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumanns book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by F k ( B ), one of its free groups of finite rank; and if so, if F n ( B ) is residually a k -generator group for all n ≧ k . (Here, as in the sequel, all unexplained notation follows [7].)

Some symmetric varieties of groups

Let ( n, σ, d ) denote the variety of all groups defined by the left-normed commutator identity [ x 1 , …, x n ] = [ x 1σ , …, x nσ ] d , where σ is a non-identity permutation of {1, …, n }, and d is an integer, possibly negative. It is shown that ( n, σ, d ) is nilpotent-by-nilpotent if σ ≠ (1, 2), abelian by nilpotent if n > 2, n σ ≠ n , and nilpotent of class at most n + 1 if {1, 2} ≠ {1σ, 2σ}. This improves on a result of E.B. Kikodze that ( n , σ, 1) is locally soluble and if {1, 2} ≠ {1σ, 2σ} is locally nilpotent.

UNITS OF THE INTEGRAL GROUP RING OF THE INFINITE DIHEDRAL GROUP

Let ZG be the integral group ring of the infinite dihedral group. It is shown that any torsion unit of ZG, having augmentation one, is conjugate in {JG to an element of G. This result is applied to deduce a result of Wallace that any normalized automorphism of ZG is composed of an automorphism of G and a conjugation by a suitable element of {JG.