Primitive group of permutations

primitive permutation group

A 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). 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 $ 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 of a primitive permutation group is transitive. Every transitive permutation group can be imbedded in a multiple 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. [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

[1]	P. Cameron, "Finite permutation groups and finite simple groups" Bull. London Math. Soc. , 13 (1981) pp. 1–22
[2]	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
[3]	H. Wielandt, "Finite permutation groups" , Acad. Press (1968) (Translated from German)
[4]	B.A. Pogorelov, "Primitive permutation groups of small degree" , VI All-Union Symp. Group Theory , Kiev (1980) pp. 146–157; 222 (In Russian)
[5]	O.Yu. Shmidt, "Abstract theory of groups" , Freeman (1966) (Translated from Russian)