Zassenhaus group

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This article is about a term related to the Classification of finite simple groups

Definition

A Zassenhaus group is a group that possesses a faithful doubly transitive group action with the property that there is no non-identity element that fixes three distinct cosets.

Importance

Zassenhaus groups arise naturally as the collineation groups of affine planes over the finite field of two elements. Here is the loose explanation:

  • The automorphisms of the affine plane over a finite field acts transitively on the set of points
  • Further, they act transitively on pairs of points: by first changing the line, then dilating by a suitable factor
  • However, if an automorphism fixes three non-collinear points, then it fixes the whole plane because of the constraint of preserving collinearity.

When the finite field has only two elements, every line has only two points, so any three distinct points fixed by an automorphism must be non-collinear, forcing that automorphism to be the identity automorphism.