New Lie products for groups and their automorphisms

We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This process uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.

The special linear group for nonassociative rings

We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

Lie automorphic loops under half-automorphisms

Automorphic loops or [Formula: see text]-loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring [Formula: see text] we can define an operation [Formula: see text] such that [Formula: see text] is an [Formula: see text]-loop. We call it Lie automorphic loop. A half-isomorphism [Formula: see text] between multiplicative systems [Formula: see text] and [Formula: see text] is a bijection from [Formula: see text] onto [Formula: see text] such that [Formula: see text] for any [Formula: see text]. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if [Formula: see text] is a group then [Formula: see text] is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism.

Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$p>r, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.

Derivations of differentially semiprime rings

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

The Wells exact sequence for the automorphism group of a Lie ring extension

We give an explicit description of the Wells map for the automorphism group of a Lie ring extension. Using this map, we construct an exact sequence for the automorphism group of a Lie ring extension similar to that for group extensions.

Derivations in differentially prime rings

Earlier properties of Lie rings [Formula: see text] of derivations in commutative differentially prime rings [Formula: see text] was investigated by many authors. We study Lie rings [Formula: see text] in the non-commutative case and shown that if [Formula: see text] is a [Formula: see text]-prime ring of characteristic [Formula: see text], then [Formula: see text] is a prime Lie ring or [Formula: see text] is a commutative ring.

Rings with simple Lie rings of Lie and Jordan derivations

Let [Formula: see text] be an associative ring. We characterize rings [Formula: see text] with simple Lie ring [Formula: see text] of all Lie derivations, reduced noncommutative Noetherian ring [Formula: see text] with the simple Lie ring [Formula: see text] of all derivations and obtain some properties of [Formula: see text]-torsion-free rings [Formula: see text] with the simple Lie ring [Formula: see text] of all Jordan derivations.

Dimension quotients of metabelian Lie rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].