Formally a permutation representation is any homomorphism from a group G to some symmetric group S(X); however, the number of such homomorphisms is infinite so the representations listed are "minimal" representations (minimal here does not mean anything formal; in particular, it does not mean irreducible.) Only the faithful transitive representations are listed. This is equivalent to listing the faithful transitive group actions.

The subject of Representation Theory generally does not involve permutation representations but instead studies linear representations. On the surface the only difference is replacing S(X) with the general linear group GL(V) where V is some vector space. At a philosophic level linear representations are more useful as they provide a module structure with which to investigate the groups. However

Given a permutation representation into S(X), the degree of the representation is the size of the set X. For linear representations into GL(V) the degree is the dimension of the vector space V.

Faithful representations are permutation representations (homomorphisms from a group G to a symmetric group S(X)) with a trivial kernel. In terms of group actions, this means the only element that leaves all points fixed is the trivial element.

The important consequence of faithful representations is that the original group is embedded as a subgroup of the symmetric group. This way we begin to talk about the group as a permutation group rather than a generic group defined in some alternate way.

Every group action can be reduced to a faithful group action by simply quotienting by the kernel. In practice most mathematicians take the term group action to actually mean faithful group action and thus also representation to be faithful representation.

Transitive representations are permutation representations whose image is a transitive group. The best way to understand this is to describe the representation in terms of group actions. Then a transitive representation is a group action which is transitive.

A group action is transitive if given any two points, there exists a group element that acts by moving the first to the second.

The importance of transitive actions/representations/groups is two fold:

• Any group action that is not transitive has severl oribits on which the group does act transitively. Thus the action can be broken up into transitive actions.
• The second and most important out come is that all the point stablizers are conjugate. In particular, they are all isomorphic. Thus it is sufficient to list only one stablizer with its generators and immediately the rest are known.

"The" stablizer of a group action is a misnomer. Given an arbitrary group action, different points will have different stablizers. Formally a stablizer of a point is the set of all elements of the group that fix this point. In the event that the group action is transitive, then all the stablizers are isomorphic, and indeed conjugate. Since most group actions are assumed to be transitive, it is common to declare "the" stablizer as here there is no ambiguity brought by changing the point in question.

Of particular interests are stablizers of transitive actions which are maximal subgroups. If such a stablizer exists, then the group action is primitive.

Primitive Actions are the smallest units of a group action. Only particular groups are primitive. It is important to emphasize the term primitive applies only to permutation groups (that is a particular group action.) Thus a group may be primitive under some representations but not under all.

A primitivity block is a subset of the set on which the group acts in which either the group moves the elements within the subset one to another, or it swaps all the elements of the block with the elements of another block. This can be thought of as an orbit in a group action except that on top of the action within the "orbit" we can also act to interchange equally sized "orbits" (blocks). The blocks partition the set into what is called a (imprimitivity) block system.

A group action is primitive if it has exactly two block systems of imprimitivity. Every group action has at least two block systems of imprimitivity which correspond to the two trivial partitions: first, the partition of every element into on equivalence class, and second the partition where every element is in its own class.

Note: it is common for fluent mathematicians to say "a simple group has no normal subgroups" by which they mean they have no "interesting" normal subgroups, but only the trivial ones. Likewsie, we often say a primitive group/action is one with no blocks by which we mean no "interesting" blocks. Only the interesting (non-trivial) blocks of an action are ever listed.

One of the simplest ways to detect primitivity is with stablizers. A transitive permutation group is primitive (that is, a faithful transitive group action is primitive) if and only if the stablizer is a maximal subgroup. In particular if the stablizer has prime index. Since the index of a stablizer is equal to the size of the orbit, it follows that a faithful transitive action on a prime order set is necessarily a primitive action.