Zassenhaus group - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Hans Julius Zassenhaus ( 28 May 1912 &ndash 21 November 1991) was a German Mathematician, known for work in many parts of In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with

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. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of 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.

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. In Mathematics, a Frobenius group is a transitive Permutation group on a Finite set, such that no non-trivial elementfixes more than one point

Examples

We let q = p^f be a power of a prime p, and write F_q for the finite field of order q. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements 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 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. In Group theory, the phrase Suzuki group refers to The Suzuki sporadic group. 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.

Definition

Examples

Further reading