Examples
We let q = pf be a power of a prime p, and write Fq for the finite field of order q. Suzuki proved that any Zassenhaus group is of one of the following four types:
- The projective special linear group PSL2(Fq) for q > 3 odd, acting on the q + 1 points of the projective line. It has order (q + 1)q(q − 1)/2.
- The projective general linear group PGL2(Fq) for q > 3. It has order (q + 1)q(q − 1).
- A certain group containing PSL2(Fq) with index 2, for q an odd square. It has order (q + 1)q(q − 1).
- The Suzuki group Suz(Fq) for q a power of 2 that is at least 8 and not a square. The order is (q2 + 1)q2(q − 1)
The degree of these groups is q + 1 in the first three cases, q2 + 1 in the last case.
Read more about this topic: Zassenhaus Group
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