Primitive permutation group

In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.

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 and the action is trivial; otherwise, the condition that G preserves no nontrivial partition implies that G is transitive. This is because for non-transitive actions either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case any nontrivial partition of X (which exists for |X|≥3) is preserved by G.

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.[1]

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:

Degree	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	OEIS
Number	1	2	2	5	4	7	7	11	9	8	6	9	4	6	22	10	4	8	4	9	4	7	5	A000019

There are a 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.

Examples

• Consider the symmetric group ${\displaystyle S_{3}}$ acting on the set ${\displaystyle X=\{1,2,3\}}$ and the permutation

${\displaystyle \eta ={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}.}$

Both ${\displaystyle S_{3}}$ and the group generated by ${\displaystyle \eta }$ are primitive.

• Now consider the symmetric group ${\displaystyle S_{4}}$ acting on the set ${\displaystyle \{1,2,3,4\}}$ and the permutation

${\displaystyle \sigma ={\begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}}.}$

The group generated by ${\displaystyle \sigma }$ is not primitive, since the partition ${\displaystyle (X_{1},X_{2})}$ where ${\displaystyle X_{1}=\{1,3\}}$ and ${\displaystyle X_{2}=\{2,4\}}$ is preserved under ${\displaystyle \sigma }$, i.e. ${\displaystyle \sigma (X_{1})=X_{2}}$ and ${\displaystyle \sigma (X_{2})=X_{1}}$.

• Every transitive group of prime degree is primitive
• The symmetric group ${\displaystyle S_{n}}$ acting on the set ${\displaystyle \{1,\ldots ,n\}}$ is primitive for every n and the alternating group ${\displaystyle A_{n}}$ acting on the set ${\displaystyle \{1,\ldots ,n\}}$ is primitive for every n > 2.

See also

• Block (permutation group theory)
• Jordan's theorem (symmetric group)
• O'Nan-Scott theorem, a classification of finite primitive groups into various types

References

1. Galois' last letter: http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament

• 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.
• Todd Rowland. "Primitive Group Action". MathWorld.