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 ...
, a
permutation group
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 it ...
''G''
acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad r ...
on a non-empty finite set ''X'' is called primitive if ''G'' acts
transitively on ''X'' and the only
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
the ''G''-action preserves are the trivial partitions into either a single set or into , ''X'', singleton sets. Otherwise, if ''G'' is transitive and ''G'' does preserve a nontrivial partition, ''G'' is called imprimitive.
While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
acting on a 2-element set. This is because for a non-transitive action, either the
orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of ''G'' form a nontrivial partition preserved by ''G'', or the group action is trivial, in which case ''all'' nontrivial partitions of ''X'' (which exists for , ''X'', ≥ 3) are preserved by ''G''.
This terminology was introduced by
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
in his last letter, in which he used the French term ''équation primitive'' for an equation whose
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is primitive.
Properties
In the same letter in which he introduced the term "primitive", Galois stated the following theorem:
[Galois used a different terminology, because most of the terminology in this statement was introduced afterwards, partly for clarifying the concepts introduced by Galois.]If ''G'' is a primitive solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
acting on a finite set ''X'', then the order of ''X'' is a power of a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p''. Further, ''X'' may be identified with an affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
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, and ''G'' acts on ''X'' as a subgroup of the affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Relat ...
.
If the set ''X'' on which ''G'' acts is finite, its cardinality is called the ''degree'' of ''G''.
A corollary of this result of Galois is that, if is an odd prime number, then the order of a solvable transitive group of degree is a divisor of
In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by must be a divisor of ), and
is the cardinality of the affine group of an affine space with elements.
It follows that, if is a prime number greater than 3, the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
and the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
of degree are not solvable, since their order are greater than
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
results from this and the fact that there are polynomials with a symmetric Galois group.
An equivalent definition of primitivity relies on the fact that every transitive action of a group ''G'' is isomorphic to an action arising from the canonical action of ''G'' on the set ''G''/''H'' of
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s for ''H'' a subgroup of ''G''. A group action is primitive if it is isomorphic to ''G''/''H'' for a
''maximal'' subgroup ''H'' of ''G'', and imprimitive otherwise (that is, if there is a proper subgroup ''K'' of ''G'' of which ''H'' is a proper subgroup). These imprimitive actions are examples of
induced representation
In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...
s.
The numbers of primitive groups of small degree were stated by
Robert Carmichael
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician.
Biography
Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was s ...
in 1937:
There are a large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and
alternating group, are subgroups of the
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Relat ...
on the 4-dimensional space over the 2-element
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 ...
.
Examples
* Consider the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
acting on the set
and the permutation
:
Both
and the group generated by
are primitive.
* Now consider the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
acting on the set
and the permutation
:
The group generated by
is not primitive, since the partition
where
and
is preserved under
, i.e.
and
.
* Every transitive group of prime degree is primitive
* The
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
acting on the set
is primitive for every ''n'' and the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
acting on the set
is primitive for every ''n'' > 2.
See also
*
Block (permutation group theory)
In mathematics and group theory, a block system for the action of a group ''G'' on a set ''X'' is a partition of ''X'' that is ''G''-invariant. In terms of the associated equivalence relation on ''X'', ''G''-invariance means that
:''x'' ~ ''y'' ...
*
Jordan's theorem (symmetric group) In finite group theory, Jordan's theorem states that if a primitive permutation group ''G'' is a subgroup of the symmetric group ''S'n'' and contains a ''p''- cycle for some prime number ''p'' < ''n'' − 2, then ''G'' is either the whole s ...
*
O'Nan–Scott theorem In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric ...
, a classification of finite primitive groups into various types
References
*
Roney-Dougal, Colva M. ''The primitive permutation groups of degree less than 2500'',
Journal of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...
292 (2005), no. 1, 154–183.
* Th
GAP
* Carmichael, Robert D., ''Introduction to the Theory of Groups of Finite Order.'' Ginn, Boston, 1937. Reprinted by Dover Publications, New York, 1956.
*{{MathWorld , author=Todd Rowland , title=Primitive Group Action , urlname=PrimitiveGroupAction
Permutation groups
Integer sequences