Geometric Characterizations in Finite Group Theory
Over the past several years one may be able to observe an increasing trend to use purely geometric arguments in the proofs of theorems in the theory of finite groups. The idea is a simple one. In the course of proving a theorem about finite groups, one displays some geometric configuration built out of a finite group G. He then proceeds to characterize the known configuration as being some very familiar geometric object. Because of this, the group G is a subgroup of the group of automorphisms of the geometric object, and this can frequently be used to characterize the group G. A good illustration of this principle would be the Suzuki-O’Nan characterization of the three dimensional projective unitary groups over a finite field by the centralizer of an involution.
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