What groups are Lie groups?

What are all the groups of Lie type? Do they have real world examples?

• Answer:

  You really want all of them? If there are any jungles in mathematics this is surely one of them. As mentioned in his comments, there isn't a simple catch-all description of all the groups of Lie type. At the same time, there's a fairly concrete description that covers many of the important ones and isn't too far from the whole truth: take some matrix group like SL_n or SO_n or SU_n or PSL_n, and plug in values from some finite field. The result is a finite group of Lie type. For example: \mbox{SL}_2(F) is the group of 2x2 matrices of determinant 1 with entries in some field F. If we choose this field to be finite, say for example we choose the smallest possible field having two elements 0 and 1, we get a finite group - in this case, all 2x2 invertible matrices with entries that are the "bits" 0 and 1 and "invertible" refers to arithmetic modulo 2. I'm never sure what people mean by "real world examples". The finite group \mbox{PSL}_2(7) with 168 elements can be realized as the symmetry group of the Klein quartic and many other objects besides. The Klein quartic has been represented (approximately) as a sculpture which is about as real life as you can get, I presume.