A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order pn with a given fixed coexponent, is independent of n, for p and n sufficiently large. Explicit formulae are calculated in the case of coexponent 3.

We construct a one-dimensional uniquely ergodic cellular automaton which is not nilpotent. This automaton can perform asymptotically infinitely sparse computation, which nevertheless never disappears completely. The construction builds on the self-simulating automaton of Gács. We also prove related results of dynamical and computational nature, including the undecidability of unique ergodicity,...

In this paper, we present various results on arcs in projective threedimensional Hjelmslev spaces over finite chain rings of nilpotency index 2. A table is given containing exact values and bounds for projective arcs in the geometries over the two chain rings with four elements.

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T .

We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.

A relationship between two natural dynamical systems on groups is established thereby giving a new characterization of Engel elements. Using this connection, various closure properties for the sets of left and right Engel elements are established. Some other dynamical systems and their relationship with Engel elements and nilpotency for finite groups are also considered.

In this paper, we prove the p-nilpotency of a finite group with assumption that some subgroups of Sylow subgroup are weakly s-semipermutable subgroups in the normalizer of Sylow subgroups. Our results unify and generalize some earlier results. Mathematics Subject Classification: 20D10, 20D15

In this paper we continue developing the theory of symplectic alternating algebras that was started in [3]. We focus on nilpotency, solubility and nil-algebras. We show in particular that symplectic alternating nil-2 algebras are always nilpotent and classify all nil-algebras of dimension up to 8.

Felix and Murillo [5] introduced the group AutΩ(X) of self-maps f of X, which satisfy Ωf = 1ΩX , and proved that the group is nilpotent with the order of nilpotency bounded by the Lusternik-Schnirelmann category of X. In this paper we construct a spectral sequence converging to the group AutΩ(X) and derive several interesting consequences.