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- Primitive_permutation_group abstract "In mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial 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.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:Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and 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.".
- Primitive_permutation_group wikiPageExternalLink www.gap-system.org.
- Primitive_permutation_group wikiPageExternalLink prim.html.
- Primitive_permutation_group wikiPageID "2739252".
- Primitive_permutation_group wikiPageRevisionID "562208648".
- Primitive_permutation_group author "Todd Rowland".
- Primitive_permutation_group hasPhotoCollection Primitive_permutation_group.
- Primitive_permutation_group title "Primitive Group Action".
- Primitive_permutation_group urlname "PrimitiveGroupAction".
- Primitive_permutation_group subject Category:Integer_sequences.
- Primitive_permutation_group subject Category:Permutation_groups.
- Primitive_permutation_group type Abstraction100002137.
- Primitive_permutation_group type Arrangement107938773.
- Primitive_permutation_group type Group100031264.
- Primitive_permutation_group type IntegerSequences.
- Primitive_permutation_group type Ordering108456993.
- Primitive_permutation_group type PermutationGroups.
- Primitive_permutation_group type Sequence108459252.
- Primitive_permutation_group type Series108457976.
- Primitive_permutation_group comment "In mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X. Otherwise, if G does preserve a nontrivial partition, G is called imprimitive.".
- Primitive_permutation_group label "Primitive permutation group".
- Primitive_permutation_group sameAs m.0803zl.
- Primitive_permutation_group sameAs Q7243579.
- Primitive_permutation_group sameAs Q7243579.
- Primitive_permutation_group sameAs Primitive_permutation_group.
- Primitive_permutation_group wasDerivedFrom Primitive_permutation_group?oldid=562208648.
- Primitive_permutation_group isPrimaryTopicOf Primitive_permutation_group.