Primitive permutation group

In the same letter in which he introduced the term "primitive", Galois stated the following theorem:[2]

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. Further, 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.

If the set X on which G acts is finite, its cardinality is called the degree of G.

A corollary of this result of Galois is that, if p is an odd prime number, then the order of a solvable transitive group of degree p is a divisor of ${\displaystyle p(p-1).}$ In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by G must be a divisor of p), and ${\displaystyle p(p-1)}$ is the cardinality of the affine group of an affine space with p elements.

It follows that, if p is a prime number greater than 3, the symmetric group and the alternating group of degree p are not solvable, since their order are greater than ${\displaystyle p(p-1).}$ Abel–Ruffini theorem results from this and the fact that there are polynomials with a symmetric Galois group.

An equivalent definition of primitivity relies on the fact that every transitive action of a group G is isomorphic to an action arising from the canonical action of G on the set G/H of cosets for H a subgroup of G. A group action is primitive if it is isomorphic to G/H for a maximal subgroup H of G, and imprimitive otherwise (that is, if there is a proper subgroup K of G of which H is a proper subgroup). These imprimitive actions are examples of induced representations.

The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:

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.

• 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.

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

• 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.