Primitive permutation group
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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 Robert Carmichael in 1937:
| Degree | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Number | 1 | 2 | 2 | 5 | 4 | 7 | 7 | 11 | 9 | 8 | 6 | 9 | 4 | 6 | 22 | 10 | 4 | 8 | 4 |
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.
The number of primitive permutation groups of degree n, for n = 0, 1, … , is recorded as sequence A000019 in the On-Line Encyclopedia of Integer Sequences.
[edit] See also
[edit] References
- Roney-Dougal, Colva M. The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005), no. 1, 154–183.
- The GAP Data Library "Primitive Permutation Groups".
- Carmichael, Robert D., Introduction to the Theory of Groups of Finite Order. Ginn, Boston, 1937. Reprinted by Dover Publications, New York, 1956.
- Rowland, Todd; Primitive Group Action. MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. [1]
