Finite groups and geometries: a view on the present state and on the future

The model: groups of Lie-Chevalley type and buildingsThis paper is not the presentation of a completed theory but rather a report on a search progressing as in the natural sciences in order to better understand the relationship between groups and incidence geometry, in some future sought-after theory Τ. The search is based on assumptions and on wishes some of which are time-dependent, variations being forced, in particular, by the search itself.A major historical reference for this subject is, needless to say, Klein's Erlangen Programme. Klein's views were raised to a powerful theory thanks to the geometric interpretation of the simple Lie groups due to Tits (see for instance), particularly his theory of buildings and of groups with a BN-pair (or Tits systems). Let us briefly recall some striking features of this.Let G be a group of Lie-Chevalley type of rank r, denned over GF(q), q = pn, p prime. Let Xr denote the Dynkin diagram of G. To these data corresponds a unique thick building B(G) of rank r over the Coxeter diagram Xr (assuming we forget arrows provided by the Dynkin diagram). It turns out that B(G) can be constructed in a uniform way for all G, from a fixed p-Sylow subgroup U of G, its normalizer NG(U) and the r maximal subgroups of G containing NG(U).

Social tourism today: Stakeholders, and supply and demand factors

As highlighted in the previous chapter, the definitions and consequently the expression of social tourism have developed and changed since its inception in the 19th century. In post-modern times there has been a significant evolution of the needs, the expectations and the possibilities (or opportunities) for holidaymaking and travel in general for the majority of people in Europe. Socio-political, economic and technological developments have forged a new context for tourism and created new travel opportunities (see Chapter 6). While the numbers of tourism trips have grown steadily over time, tourism participation levels in Europe have largely stabilised: there are still a number of groups in contemporary society who are excluded from tourism. Social tourism has adapted to societal changes and has changed its focus from factory workers and manual labourers towards the current main four target groups.

Resolved Torsional Splittings in the ν18 and ν19- Bands of Propene

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