K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

The influence of gender in the direction of Spanish cooperative societies on profitability and indebtedness: an empirical analysis

In this paper the claim for the market for a new business management to ensure the presence of women in decision -making to respond to new social needs addressed. Thus, this paper analyzes the influence of gender diversity of the directors on the profitability and the level of debt for a sample of 5,199 Spanish cooperatives. Unlike capitalist societies, these organizations have a number of peculiarities in their government, and that the partners are themselves major time, agents and customers. The study focuses on the Spanish context, where there is an open debate on the importance of women's business management, as in other countries, driven by the proliferation of legislation on gender equality, being, in addition, Spain, the pioneer in having specific legislation on Social Economy. The results show that cooperatives with greater female representation in theirs Boards have higher profitability. On the other hand, those Boards with a higher percentage of women show a lower level of indebtedness.