Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G and let Rn(g) be the subgroup generated by all the right Engel values [g,nx] over x∈G. In the case when G is soluble we prove that if, for some n, the Fitting height of Rn(g) is equal to k, then g belongs to the (k+1)th Fitting subgroup Fk+1(G). For nonsoluble G, it is proved that if, for some n, the generalized Fitting height of Rn(g) is equal to k, then g belongs to the generalized Fitting subgroup F∗f(k,m)(G) with f(k,m) depending only on k and m, where |g| is the product of m primes counting multiplicities. It is also proved that if, for some n, the nonsoluble length of Rn(g) is equal to k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. Earlier similar generalizations of Baer's theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups

Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page

Reference frames and rigid motions in relativity: Applications

The concept of rigid reference frame and of constricted spatial metric, given in the previous work [\emph{Class. Quantum Grav.} {\bf 21}, 3067,(2004)] are here applied to some specific space-times: In particular, the rigid rotating disc with constant angular velocity in Minkowski space-time is analyzed, a new approach to the Ehrenfest paradox is given as well as a new explanation of the Sagnac effect. Finally the anisotropy of the speed of light and its measurable consequences in a reference frame co-moving with the Earth are discussed.Comment: 13 pages, 1 figur

The relativistic Sagnac Effect: two derivations

The phase shift due to the Sagnac Effect, for relativistic matter and electromagnetic beams, counter-propagating in a rotating interferometer, is deduced using two different approaches. From one hand, we show that the relativistic law of velocity addition leads to the well known Sagnac time difference, which is the same independently of the physical nature of the interfering beams, evidencing in this way the universality of the effect. Another derivation is based on a formal analogy with the phase shift induced by the magnetic potential for charged particles travelling in a region where a constant vector potential is present: this is the so called Aharonov-Bohm effect. Both derivations are carried out in a fully relativistic context, using a suitable 1+3 splitting that allows us to recognize and define the space where electromagnetic and matter waves propagate: this is an extended 3-space, which we call "relative space". It is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the 'physical space of the rotating platform': the geometry of this space turns out to be non Euclidean, according to Einstein's early intuition.Comment: 49 pages, LaTeX, 3 EPS figures. Revised (final) version, minor corrections; to appear in "Relativity in Rotating Frames", ed. G. Rizzi and M.L. Ruggiero, Kluwer Academic Publishers, Dordrecht, (2003). See also http://digilander.libero.it/solciclo