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Primitive permutation groups of prime power degree are known to be affine type, almost simple type, and product action type. At the present stage finding an explicit classification of primitive groups of affine type seems untractable, while the product action type can usually be reduced to almost simple type. In this paper, we present a short survey of the development of primitive groups of prime power degree, together with a brief description on such groups.

Transitive groups, in particular, primitive groups, of special degrees have received much attention in the literature. As early as 1832, Galois showed that the projective linear groups

One of the pioneers of investigating primitive permutation groups of prime power degree is G. Jones in 1975, in his Ph.D. thesis. At that time, an explicit description on such groups was not available because the classification of the finite simple groups had not been completed. In 1976, Praeger also studied primitive permutation groups of prime power degree [

In 2002, Dobson and Witte [

In recent years, the problem of determining transitive (or primitive) permutation groups of special degree is closely related to some important combinatorial problems, such as the problem of classifying symmetric graphs, symmetric Cayley graphs, edge-transitive graphs, and half-arc-transitive graphs, of specific degree.

In this paper, we take a simple retrospect on the analyzing process of the primitive permutation groups of prime power degree and present a brief description of primitive permutation groups of degree

Our description of primitive permutation groups of prime power degree is given in the following theorem.

Let

The subgroups of

Let

Let

For a set

Let

The primitivity of a transitive permutation group may be characterized by the maximality of its point stabilizer, that is, the following well-known result.

A transitive permutation group

The structure of finite primitive groups is characterized by the famous O’Nan-Scott Theorem (see [

HA (holomorph affine):

AS (almost simple):

SD (simple diagonal):

PA (product action):

TW (twisted wreath product):

In the above description,

Let

Let

It is clear that if

In [

Let

As a result, we have the following.

Let

Let

We point out that if

In [

The socle of a finite doubly transitive group is either a regular elementary abelian

It follows from the above result that a doubly transitive group is of type HA or AS.

Let

If

Find out all irreducible subgroups of

If the type of

Let

Transitive groups of prime degree are known for a very long time and are given as follows.

Let

Primitive groups of large prime power degrees sometimes appear as the wreath product of primitive groups of small prime power degrees, in the product action. The spirit is the following well-known result.

Suppose that

It follows from the above result that, for

The wreath product

In this section, we deal with the case of

Let

It is clear that there are no simply primitive groups of degree 8, so

Now, we prove Theorem

Let

If

Assume now that

This completes the proof of Theorem

The result of Theorem

Study imprimitive groups of degree

The authors declare that there is no conflict of interests regarding the publication of the paper.

This work is supported by the NNSF (11161058) and YNSF (2011FZ087).