Coherence bandwidth

2014-02-03T14:55:02Z

134.191.221.72:

In [[mathematics]], a [[permutation group]] ''G'' [[group action|acting]] on a set ''X'' is called '''primitive''' if ''G'' acts [[Group_action#Types_of_actions|transitively]] on ''X'' and ''G'' preserves no nontrivial [[Partition_of_a_set|partition]] of ''X''. Otherwise, if ''G'' does preserve a nontrivial partition, ''G'' is called '''imprimitive'''.

This terminology has been introduced in his last letter by [[Évariste Galois]] who called (in French) ''equation primitive'' an equation whose [[Galois group]] is primitive.<ref> Galois' last letter: http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament</ref>

In the same letter he stated also the following theorem.

If ''G'' is a primitive [[solvable group]] acting on a finite set ''X'', then the order of ''X'' is a power of a [[prime number]] ''p'', ''X'' may be identified with an [[affine space]] over the [[finite field]] with ''p'' elements and ''G'' acts on ''X'' as a subgroup of the [[affine group]].

An imprimitive permutation group is an example of an [[induced representation]]; examples include [[coset]] representations ''G''/''H'' in cases where ''H'' is not a [[maximal subgroup]]. When ''H'' is maximal, the coset representation is primitive.

If the set ''X'' is finite, its cardinality is called the "degree" of ''G''.
The numbers of primitive groups of small degree were stated by [[Robert Carmichael]] in 1937:

{| class="wikitable"
| Degree || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || [[OEIS]]
|-
| Number || 1 || 2 || 2 || 5 || 4 || 7 || 7 || 11 || 9 || 8 || 6 || 9 || 4 || 6 || 22 || 10 || 4 || 8 || 4 || {{OEIS link|id=A000019}}
|}

Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the [[symmetric group|symmetric]] and [[alternating group|alternating]] group, are subgroups of the [[affine group]] on the 4-dimensional space over the 2-element [[finite field]].

While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when ''X'' is a 2-element set; otherwise, the condition that ''G'' preserves no nontrivial partition implies that ''G'' is transitive.

== Examples ==
* Consider the [[symmetric group]] <math>S_3</math> acting on the set <math>X=\{1,2,3\}</math> and the permutation
: <math>\eta=\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \end{pmatrix}.</math>
Both <math>S_3</math> and the group generated by <math>\eta</math> are primitive.
* Now consider the [[symmetric group]] <math>S_4</math> acting on the set <math>\{1,2,3,4\}</math> and the permutation
: <math>\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1 \end{pmatrix}.</math>
The group generated by <math>\sigma</math> is not primitive, since the partition <math>(X_1, X_2)</math> where <math>X_1 = \{1,3\}</math> and <math>X_2 = \{2,4\}</math> is preserved under <math>\sigma</math>, i.e. <math>\sigma(X_1) = X_2</math> and <math>\sigma(X_2)=X_1</math>.
* Every transitive group of prime degree is primitive
* The [[symmetric group]] <math>S_n</math> acting on the set <math>\{1,\ldots,n\}</math> is primitive for every ''n'' and the [[alternating group]] <math>A_n</math> acting on the set <math>\{1,\ldots,n\}</math> is primitive for every ''n'' > 2.

== See also ==
* [[Block (permutation group theory)]]
* [[Jordan's theorem (symmetric group)]]

== References ==
{{Reflist}}
* Roney-Dougal, Colva M. ''The primitive permutation groups of degree less than 2500'', [[Journal of Algebra]] 292 (2005), no. 1, 154–183.
* The [http://www.gap-system.org GAP] [http://www.gap-system.org/Datalib/prim.html Data Library "Primitive Permutation Groups"].
* 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}}

[[Category:Permutation groups]]
[[Category:Integer sequences]]

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Coherence bandwidth