Primitive permutation group

In the same letter in which he introduced the term "primitive", Galois stated the following theorem:Galois used a different terminology, because most of the terminology in this statement was introduced afterwards, partly for clarifying the concepts introduced by Galois.

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.

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

It follows that, if {{mvar|p}} is a prime number greater than 3, the symmetric group and the alternating group of degree {{mvar|p}} are not solvable, since their order are greater than $p(p-1).$ 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,{{pages?|date=October 2021}} 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 $S\_3$ acting on the set $X=\backslash \{1,2,3\backslash \}$ and the permutation

: $\backslash eta=\backslash begin\{pmatrix\}$

1 & 2 & 3 \\

2 & 3 & 1 \end{pmatrix}.

Both $S\_3$ and the group generated by $\backslash eta$ are primitive.

• Now consider the symmetric group $S\_4$ acting on the set $\backslash \{1,2,3,4\backslash \}$ and the permutation

: $\backslash sigma=\backslash begin\{pmatrix\}$

1 & 2 & 3 & 4 \\

2 & 3 & 4 & 1 \end{pmatrix}.

The group generated by $\backslash sigma$ is not primitive, since the partition $(X\_1,\; X\_2)$ where $X\_1\; =\; \backslash \{1,3\backslash \}$ and $X\_2\; =\; \backslash \{2,4\backslash \}$ is preserved under $\backslash sigma$, i.e. $\backslash sigma(X\_1)\; =\; X\_2$ and $\backslash sigma(X\_2)=X\_1$.

• Every transitive group of prime degree is primitive
• The symmetric group $S\_n$ acting on the set $\backslash \{1,\backslash ldots,n\backslash \}$ is primitive for every n and the alternating group $A\_n$ acting on the set $\backslash \{1,\backslash ldots,n\backslash \}$ 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

{{Reflist}}

• Roney-Dougal, Colva M. The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005), no. 1, 154–183.
• The [http://www.gap-system.org GAP] [http://www.gap-system.org/Datalib/prim.html 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.
• {{MathWorld |author=Todd Rowland |title=Primitive Group Action |urlname=PrimitiveGroupAction}}

Category:Permutation groups

Category:Integer sequences