# Zassenhaus group

In mathematics, a **Zassenhaus group**, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.

A **Zassenhaus group** is a permutation group *G* on a finite set *X* with the following three properties:

Some authors omit the third condition that *G* has no regular normal subgroup. This condition is put in to eliminate some "degenerate" cases. The extra examples one gets by omitting it are either Frobenius groups or certain groups of degree 2^{p} and order
2^{p}(2^{p} − 1)*p* for a prime *p*, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2^{p}.

We let *q* = *p ^{f}* be a power of a prime

*p*, and write

*F*for the finite field of order

_{q}*q*. Suzuki proved that any Zassenhaus group is of one of the following four types:

The degree of these groups is *q* + 1 in the first three cases, *q*^{2} + 1 in the last case.