mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Zassenhaus group, named after

Hans Zassenhaus 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. Biography He was born in Koblenz in 1912. His father was a historian and ...

, is a certain sort of

doubly transitive permutation group A group G acts 2-transitively on a set S if it acts transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq ...

very closely related to rank-1

groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...

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 abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

. ("Regular" means that non-trivial elements do not fix any points of ''X''; compare

free action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...

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

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

pings 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 In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

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

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...

''PSL''₂(''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 In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...

''PGL''₂(''F''_''q'') for ''q'' > 3. It has order (''q'' + 1)''q''(''q'' − 1). *A certain group containing ''PSL''₂(''F''_''q'') with

index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...

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 (''q''² + 1)''q''²(''q'' − 1) The degree of these groups is ''q'' + 1 in the first three cases, ''q''² + 1 in the last case.

Definition

Examples

Further reading