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In mathematics, a Zassenhaus group, named after Hans Julius 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:

The degree of a Zassenhaus group is the number of elements of X.

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 2p and order 2p(2p − 1)p for a prime p, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2p.

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 degree of these groups is q + 1 in the first three cases, q2 + 1 in the last case.

Further reading

Finite Groups III (Grundlehren Der Mathematischen Wissenschaften Series, Vol 243) by B. Huppert, N. Blackburn, ISBN 0-387-10633-2

Mathematics Encyclopedia

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