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In mathematics, a permutation group G acting on a set X is called primitive if G preserves no nontrivial partition of X. In the other case, G is imprimitive. 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 R. D. Carmichael in 1937:
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