# 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. ). 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.

How to Cite This Entry:
Primitive group of permutations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_group_of_permutations&oldid=48285
This article was adapted from an original article by L.A. Kaluzhnin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article