Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. In particular, for each integer r > 1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups...

A nonassociative algebra is nilpotent if there is some n such that the product of any n elements, no matter how they are associated, is zero. Several related, but more general, notions are left nilpotency, solvability, local nilpotency, and nillity. First the complexity of several decision problems for these properties is examined. In finite-dimensional algebras over a finite field it is shown ...

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. As of now, no other examples are known. We construct many such loops from groups of nilpotency class two by replacing the product xy with xyh in certain positions, where h is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form. c ...

By T. Kepka and M. Niemenmaa if the inner mapping group of a finite loop Q is abelian, then the loop Q is centrally nilpotent. For a long time there was no example of a nilpotency degree greater than two. In the nineties T. Kepka raised the following problem: whether every finite loop with abelian inner mapping group is centrally nilpotent of class at most two. For many years the prevailing opi...

In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal p-subgroup ∆(G) of the automorphism group, Aut(G) of a bounded (or finite) abelian p-group G. This is the first part of two papers devoted to compute the nilpotency class of ∆(G) using formulas, and algorithms that work in almost all groups. Here, we prove that for p ≥ 3 t...

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety. Surveys of the work on the characterization of the fundamental groups of smooth projective varieties and Kähler manifolds (see [1],[3], [9]) indicate tha...

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Csörgő type. In order to obtain small loops of Csörgő type, we expand our programme from Explicit constructions of loops with commuting inner mappings, European J. Combin. 29 (2008), 1662–1681, and analyze the foll...

Here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. Also we find integers $n$ for which, these groups are $n$-central.

We obtain an explicit bound for the nilpotency class of n-Engel Lie algebras of characteristic zero. 1991 Mathematics Subject Classification (Amer. Math. Soc.): 17B30.