In
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 ...
, especially in areas of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
and
finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
, the list of transitive finite linear groups is an important classification of certain highly symmetric
actions
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fi ...
of
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s on
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s.
The
solvable finite
2-transitive 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 ...
s were classified by
Bertram Huppert
Bertram Huppert (born 22 October 1927 in Worms, Germany) is a German mathematician specializing in group theory and the representation theory of finite groups. His ''Endliche Gruppen'' ( finite groups) is an influential textbook in group theory, ...
. The
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
made possible the complete classification of finite
doubly transitive 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 ...
permutation groups
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
. This is a result by
Christoph Hering. A finite 2-transitive group has a
socle that is either a vector space over a
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 ...
or a non-abelian primitive
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
; groups of the latter kind are almost simple groups and described elsewhere. This article provides a complete list of the finite 2-transitive groups whose socle is
elementary abelian
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
.
Let
be a prime, and
a subgroup of the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
acting transitively on the nonzero vectors of the ''d''-dimensional vector space
over 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 ...
with ''p'' elements.
Infinite classes
There are four infinite classes of finite transitive linear groups.
*
*
*
*
Notice that the exceptional
group 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 ...
''G''
2(''q'') is usually constructed as the automorphism groups of the split
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s. Hence, it has a natural
representation as a subgroup of the 7-dimensional
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(7, ''q''). If ''q'' is even, then the underlying
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
polarizes to a degenerate
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
. Factoring out with the radical, one obtains an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
between O(7, ''q'') and the
symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
Sp(6, ''q''). The subgroup of Sp(6, ''q'') which corresponds to ''G''
2(''q'')′ is transitive.
In fact, for ''q''>2, the group ''G''
2(''q'') = ''G''
2(''q'')′ is simple. If ''q''=2 then ''G''
2(2)′ ≅ PSU(3,3) is simple with index 2 in ''G''
2(2).
Sporadic finite transitive linear groups
These groups are usually classified by some typical
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 ...
, this normal subgroup is denoted by ''G''
0 and are written in the third column of the table. The notation 2
1+4− stands for the
extraspecial group In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime ''p'' and positive integer ''n'' there are exactly two (up to isomorphism) extraspeci ...
of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).
All but one of the sporadic transitive linear groups
yield a primitive permutation group
of degree at most 2499. In the computer algebra programs
GAP and
MAGMA
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, these groups can be accessed with the command
PrimitiveGroup(p^d,k);
where the number ''k'' is the ''primitive identification'' of
. This number is given in the last column of the following table.
Seven of these groups are sharply transitive; these groups were found by
Hans Zassenhaus and are also known as the multiplicative groups of the Zassenhaus
near-fields. These groups are marked by a star in the table.
This list is not explicitly contained in Hering's paper. Many books and papers give a list of these groups, some of them an incomplete one. For example, Cameron's book
misses the groups in line 11 of the table, that is, containing
as a normal subgroup.
References
{{reflist
Permutation groups