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.

An experiment in group learning technology: evaluating critical thinking in face-to-face and computer-supported seminars

Can learning quality be maintained in the face of increasing class size by the use of Computer Supported Co-operative Learning (CSCL) technologies? In particular, can Computer-Mediated Communication promote critical thinking in addition to surface information transfer? We compared face-to-face seminars with asynchronous computer conferencing in the same Information Management class. From Garrison's theory of critical thinking and Henri's critical reasoning skills, we developed two ways of evaluating critical thinking: a student questionnaire and a content analysis technique. We found evidence for critical thinking in both situations, with some subtle differences in learning style. This paper provides an overview of this work.

A content analysis method to measure critical thinking in face-to-face and computer supported group learning

This paper gives a detailed account of the content analysis method developed at Queen's University Belfast to measure critical thinking during group learning, as used in our controlled comparisons between learning in face-to-face and computer conference seminars. From Garrison's 5 stages of critical thinking, and Henri's cognitive skills needed in CMC, we have developed two research instruments: a student questionnaire and this content analysis method. The content analysis relies on identifying, within transcripts, examples of indicators of obviously critical and obviously uncritical thinking, from which several critical thinking ratios can be calculated.