In mathematics, a Zassenhaus group is a certain sort of doubly transitive permutation group. They are very closely related to rank 1 groups of Lie type. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Hans Julius Zassenhaus (28 May 1912 - 21 November 1991) was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...

Definition

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

• G is doubly transitive.

• Non-trivial elements of G fix at most 2 points.

• G has no regular normal subgroup. ("Regular" means that non-trivial elements do not fix any points of X.)

The degree of a Zassenhaus group is the number of elements of X. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way...

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.

Examples

We let q=p^f be a power of a prime p, and write F_q for the finite field of order q. Suzuki proved that any Zassenhaus group is of one of the following four types: In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

• The projective special linear group PSL₂(F_q) 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(F_q) for q>3. It has order (q+1)q(q-1).

• A certain group containing PSL₂(F_q) with index 2, for q an odd square. It has order (q+1)q(q-1).

• The Suzuki group Suz(F_q) 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.

Further reading

• Finite Groups III (Grundlehren Der Mathematischen Wissenschaften Series, Vol 243) by B. Huppert, N. Blackburn, ISBN 0387106332