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''primitive permutation group''

A [[Permutation group|permutation group]]  $  ( G, M) $
that preserves only the trivial equivalences on the set  $  M $(
i.e. equality and amorphous equivalence). For the most part, finite primitive groups are studied.

A primitive permutation group is transitive, and every  $  2 $-
transitive group is primitive (cf. [[Transitive group|Transitive group]]). Proper  $  1 $-
transitive (i.e. not  $  2 $-
transitive) permutation groups are called uniprimitive. The commutative primitive permutation groups are precisely the cyclic groups of prime order. A transitive permutation group is primitive if and only if the [[Stabilizer|stabilizer]]  $  G _ {a} $
of any  $  a \in M $
is a maximal subgroup in the group  $  G $. 
Another criterion for primitivity is based on associating with each transitive group  $  ( G, M) $
the graphs determined by the binary orbits of this group. A group  $  ( G, M) $
is primitive if and only if the graphs corresponding to non-reflexive  $  2 $-
orbits are connected. The number of  $  2 $-
orbits is called the rank of the group  $  ( G, M) $. 
The rank is 2 for doubly-transitive groups, while the rank of a uniprimitive group is at least 3.

Every non-identity [[Normal subgroup|normal subgroup]] of a primitive permutation group is transitive. Every transitive permutation group can be imbedded in a multiple [[Wreath product|wreath product]] of primitive permutation groups. (However, such a representation is not unique.)

Many questions on permutation groups reduce to the case of primitive permutation groups. All primitive permutation groups of order  $  \leq  50 $
are known (cf. [[#References|[4]]]). The relation between primitive permutations groups and finite simple groups has been much investigated.

A generalization of the notion of a primitive permutation group is that of a multiply primitive group. A permutation group  $  ( G, M) $
is called  $  k $-
fold primitive if it is  $  k $-
fold transitive and if the pointwise stabilizer of  $  ( k - 1) $
points acts primitively on the remaining points.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Cameron,   "Finite permutation groups and finite simple groups"  ''Bull. London Math. Soc.'' , '''13'''  (1981)  pp. 1–22</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Krasner,   L. Kaloujnine,   "Produit complet des groupes de permutations et problème d'extension de groupes II"  ''Acta Sci. Math. (Szeged)'' , '''14'''  (1951)  pp. 39–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Wielandt,   "Finite permutation groups" , Acad. Press  (1968)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Pogorelov,   "Primitive permutation groups of small degree" , ''VI All-Union Symp. Group Theory'' , Kiev  (1980)  pp. 146–157; 222  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.Yu. Shmidt,   "Abstract theory of groups" , Freeman  (1966)  (Translated from Russian)</TD></TR></table>