In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Zassenhaus group, named after Hans Julius Zassenhaus

Hans Julius Zassenhaus

Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra....

, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.

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 two points.

• G has no regular normal subgroup
  Normal subgroup
  In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
  
  . ("Regular" means that non-trivial elements do not fix any points of X; compare free action.)

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 group

Frobenius group

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...

s 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

Semilinear transformation

In linear algebra, particularly projective geometry, a semilinear transformation between vector spaces V and W over a field K is a function that is a linear transformation "up to a twist", hence semi-linear, where "twist" means "field automorphism of K"...

 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

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 of order q. Suzuki proved that any Zassenhaus group is of one of the following four types:

• 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
  Index of a subgroup
  In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
  
   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 0-387-10633-2