it's a nice delight for a mathematician to witness the expansion and enlargement of a thought during which he has taken a few half in the course of its early years. whilst H. Weyl coined the phrases "classical groups", most advantageous in his brain have been their connections with invariant conception, which his recognized ebook helped to restore. even though his technique in that booklet was once intentionally algebraic, his curiosity in those teams without delay derived from his pioneering examine of the precise case during which the scalars are actual or complicated numbers, the place for the 1st time he injected Topology into Lie idea. yet ever because the definition of Lie teams, the analogy among basic classical teams over finite fields and straightforward classical teams over IR or C were saw, whether the idea that of "simplicity" was once no longer really a similar in either instances. With the invention of the phenomenal easy complicated Lie algebras via Killing and E. Cartan, it was once normal to appear for corresponding teams over finite fields, and already round 1900 this was once performed by way of Dickson for the outstanding Lie algebras G and E • despite the fact that, a deep explanation for this 2 6 parallelism used to be lacking, and it's only Chevalley who, in 1955 and 1961, stumbled on that to every complicated basic Lie algebra corresponds, via a uniform method, a bunch scheme (fj over the hoop Z of integers, from which, for any box ok, can be derived a bunch (fj(K).