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- Q7243579 subject Q10036026.
- Q7243579 subject Q5519261.
- Q7243579 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 is transitive and G does preserve a nontrivial partition, G is called imprimitive. This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for 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.".
- Q7243579 wikiPageExternalLink www.gap-system.org.
- Q7243579 wikiPageExternalLink prim.html.
- Q7243579 wikiPageWikiLink Q10036026.
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- Q7243579 wikiPageWikiLink Q5519261.
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- Q7243579 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 is transitive and G does preserve a nontrivial partition, G is called imprimitive.".
- Q7243579 label "Primitive permutation group".