|
|
| Line 1: |
Line 1: |
| 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'''.
| | The author's title is Andera and she believes it sounds fairly great. Some time in the past she chose to live in Alaska and her parents reside close by. As a lady what she truly likes is fashion and she's been doing it for fairly a while. I am an invoicing officer and I'll be promoted soon.<br><br>Also visit my website; psychic readings online ([http://mybrandcp.com/xe/board_XmDx25/107997 mybrandcp.com]) |
| | |
| 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]]
| |
The author's title is Andera and she believes it sounds fairly great. Some time in the past she chose to live in Alaska and her parents reside close by. As a lady what she truly likes is fashion and she's been doing it for fairly a while. I am an invoicing officer and I'll be promoted soon.
Also visit my website; psychic readings online (mybrandcp.com)