A Note on Primitive Permutation Groups of Prime Power Degree

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.


Introduction
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 PSL 2 () have permutation representations of degree  with  = 5, 7, and 11.In 1861, Mathieu discovered his famous multiply transitive groups, including two sporadic simple groups M 11 and M 23 , of degrees 11 and 23, respectively.In 1901, Burnside [1] classified transitive groups of prime degree, showing that such groups are doubly transitive or contain a normal regular -group with  prime.As a well-known consequence of the classification of the finite simple groups (CFSG), all doubly transitive groups are known [2,Theorem 5.3], and hence all doubly transitive groups of prime degree are known.It means that, by using the CFSG, one can easily obtain Burnside's classification result.Later in 1983 Guralnick [3] studied primitive simple groups of prime power degree, in 1985 Liebeck and Saxl [4] classified primitive groups of odd degree, and in 2003 Li and Seress classified primitive groups of square-free degree [5].In [6], Li gave a list of primitive permutation groups of degree , where  = 2 ⋅ 3  , 5 ⋅ 3  , or 10 ⋅ 3  , with  a prime.
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 [7].Then, in 1979 at the Santa Cruz Conference in finite groups, M. O'Nan and L. L. Scott independently proposed a classification scheme for finite primitive groups, which became a theorem of O'Nan and Scott and finally the O'Nan-Scott Theorem; see [8].This theorem has been proved to be very important in studying finite primitive groups.Nevertheless, when dealing with primitive groups of special degrees, more work needs to be done.
In 2002, Dobson and Witte [9] determined all the transitive groups of degree  2 , with  a prime, whose Sylow -subgroup is not isomorphic to the wreath product Z  ≀ Z  .Hence, it is a natural next step to carry out a study on transitive groups of degree  3 or, more generally, of degree   , with  a prime.
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 Theorem 1.Let  be a primitive permutation group of degree   , with  primes, where  ≥ 3.Then, one of the following holds.
(a)  is doubly transitive and either

Primitive Groups of Degree 𝑝 𝑟
Let Ω be a nonempty set.Recall that the symmetric group Sym(Ω) is defined to be the group of all permutations of Ω.
A permutation group  on Ω is simply a subgroup of the symmetric group Sym(Ω), and the size |Ω| is called the degree of .If |Ω| = , then Sym(Ω) is also denoted by   .The set of the even permutations of   forms a subgroup of   , which is called the alternating group of degree  and is denoted by Let  ≤ Sym(Ω).For  ∈ Ω, we use   to denote the point stabilizer of  with respect to .The orbit of  containing  is defined to be the set The group  is called transitive if, for any two points ,  ∈ Ω, there exists an element  ∈  such that   = , which is equivalent to saying that  has only one orbit on Ω.A group  ≤ Sym(Ω) is called semiregular if   = 1 for every  ∈ Ω, and  is called regular if  is transitive and semiregular.Observe that any proper subgroup of a regular group is semiregular but not regular.
For a set Ω, let B = { 1 ,  2 , . . .,   } be a set of subsets of Ω.Then, B is said to be a partition of Ω if Ω is the disjoint union of   ; that is, Ω =  1 ∪  2 ⋅ ⋅ ⋅ ∪   , and   ∩   = 0 for  ̸ = .There are two trivial partitions of Ω: B = {Ω} and B = { 1 ,  2 , . . .,   } with |  | = 1.Let  be a transitive permutation group on Ω.A partition B of Ω is said to be invariant, if, for any  ∈  and  ∈ B,   ∈ B. In this case, B is also said to be an imprimitive partition of  on Ω.
Let  ≤ Sym(Ω) be transitive.Then,  is called primitive if it has no nontrivial -invariant partition.If B is an imprimitive partition of  on Ω, then |B| divides |Ω|, and so each transitive permutation group of prime degree is primitive.Let Ω () be the set of all -tuples of points in Ω; that is, Then, one can define an action of  on Ω () by A group  ≤ Sym(Ω) is said to be -transitive if  is transitive on Ω () .A 2-transitive group is also called doubly transitive.
It is well-known that a doubly transitive group is primitive.If  is primitive but not doubly transitive, then we say that  is simply primitive.
The primitivity of a transitive permutation group may be characterized by the maximality of its point stabilizer, that is, the following well-known result.Proposition 3. A transitive permutation group  ≤ Sym (Ω), where |Ω| ≥ 2, is primitive if and only if each of its point stabilizers is a maximal subgroup of .
The structure of finite primitive groups is characterized by the famous O'Nan-Scott Theorem (see [11, page 106] or [8]).By the O'Nan-Scott Theorem, finite primitive groups can be divided into five disjoint types, known as HA, AS, SD, PA, and TW.Thus, let  ≤ Sym(Ω) be a primitive permutation group, and let  = soc() be the socle of , that is, the product of all minimal normal subgroups of .Then, we can give a brief description of the five types of finite primitive groups as follows.PA (product action):  ≅   , where  ≥ 2,  < ,  | , and let  be a primitive group of  with soc() ≅   ; then,  is isomorphic to a subgroup of ≀Sym(/), with the product action, and |Ω| =  / .TW (twisted wreath product):  ≅   , where  ≥ 6, and  is regular on Ω, with |Ω| = ||  .Remark 4. In the above description,  is a nonabelian simple group and  is primitive of type AS or SD.Lemma 5. Let  be a primitive permutation group acting on a set Ω, where |Ω| =   ( ∈ N) for some prime .Then, the type of  is HA, AS, or PA.

HA (holomorph affine
Proof.Let  be a primitive permutation group acting on a set Ω, let  be the socle of , and let  ∈ Ω.Then, by the O'Nan-Scott Theorem, the type of  is HA, AS, SD, PA, or TW.Suppose that the type of  is SD.Then,  ≥ 2, |Ω| = || −1 =   .Since  is a nonabelian simple, this is not possible.Similarly the type of  is not TW.Thus, the type of  is HA, AS, or PA.
It is clear that if  = 1, then the type of  is HA or AS.In [3], Guralnick classified finite simple group with a subgroup index a prime power.
Theorem 6 (Guralnick,[3]).Let  be a nonabelian simple group, let  < , and let |[ : ]| =   , where  ∈ N and  is a prime.Then, one of the following holds.As a result, we have the following.We point out that if  = 2, then either  =  2  or  = PSL 2 (), such that  = 2  − 1 is a Mersenne prime.
In [12], Burnside proved the following result.
Theorem 8.The socle of a finite doubly transitive group is either a regular elementary abelian -group or a nonabelian simple group.
It follows from the above result that a doubly transitive group is of type HA or AS.
Let  be a doubly transitive group of degree   , with  a prime.If the type of  is HA, then the socle of  is abelian, which is isomorphic to an elementary abelian group Z   .In this case,  = Z   :   ≤ AΓL  (), and   is an irreducible subgroup of ΓL  () which is transitive on nonzero vectors of Z   .Such groups  have been determined by Huppert in [13] for soluble case and by Hering [10] for insoluble case.
If  is simply primitive of affine type, then the problem is very hard, and there is a long standing open problem.
Open Problem 1. Find out all irreducible subgroups of GL  () which are not transitive on the nonzero vectors of Z   .
If the type of  is AS, then  can be easily read off from Theorem 6.Note that doubly transitive groups with nonabelian socles are also listed in [2, page 8], as a result of the classification of finite simple groups.By Theorem 6 or by inspecting the list, we see that the only doubly transitive groups of degree   ( ≥ 2) are as follows.
Transitive groups of prime degree are known for a very long time and are given as follows.
Theorem 10 (see [11, page 99]).Let  be a transitive group of prime p degree.Then, one of the following statements holds.
(2)  is primitive of almost simple type, and one of the following holds: 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.
Proposition 11 (see [11, page 50]).Suppose that  and  are nontrivial permutation groups acting on the sets Γ and Δ, respectively.Then, the wreath product  =  ≀  is primitive in the product action on Ω = Δ |Γ| if and only if (i)  acts primitively but not regularly on Δ; (ii) Γ is finite and  acts transitively on Γ.
It follows from the above result that, for |Ω| =   , with  prime, we have the next corollary.Proof.Let  be a primitive permutation group acting on a set Ω, where |Ω| =   for some prime , so  is either doubly primitive or simply primitive on Ω.By Lemma 5, we know that  is of type HA, AS, or PA.Let  be the socle of .
If  is doubly primitive on Ω, then by Theorem 8, the type of  is HA or AS,  ≅ Z   or  where  is nonabelian simple.In the former case,  = Z   :   ≤ AΓL  () and   is transitive on the nonzero vectors of Z   , such subgroups are determined by Hering in [10].In the latter case, by Theorem 9, either  = A   or S   or PSL  () ≤  ≤ PΓL  (), and  =    or PSL  () where (  − 1)/( − 1) =   .Thus, (a) of Theorem 1 holds.
This completes the proof of Theorem 1.
The result of Theorem 1 gives rise to the following natural question.
Question 1. Study imprimitive groups of degree   with  a prime.
are determined by Hering in[10].
] =   .Therefore, the pair (,   ) = (, ) lies in the list of Theorem 6. Combining this list with the list of doubly transitive groups given in[2,Table on page 8], we conclude that  acts doubly transitive on Ω unless  ≅ PSU 4 (2), which has a maximal subgroup  ≅ Z 4 2 :  5 , of index 27, and  is simply primitive on Ω = [ : ].Therefore, the result is true.