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.
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 have permutation representations of degree with , and . In 1861, Mathieu discovered his famous multiply transitive groups, including two sporadic simple groups and , of degrees and , respectively. In 1901, Burnside  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  studied primitive simple groups of prime power degree, in 1985 Liebeck and Saxl  classified primitive groups of odd degree, and in 2003 Li and Seress classified primitive groups of square-free degree . In , Li gave a list of primitive permutation groups of degree , where , , or , 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 . 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 . 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  determined all the transitive groups of degree , with a prime, whose Sylow -subgroup is not isomorphic to the wreath product . Hence, it is a natural next step to carry out a study on transitive groups of degree 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 degree and present a brief description of primitive permutation groups of degree , with a prime. In the spirit of the Burnside Theorem and Dobson’s work, here we may assume that . This work forms the first part of a larger program, namely, to classify transitive permutation groups of prime power degree.
Our description of primitive permutation groups of prime power degree is given in the following theorem.
Theorem 1. Let be a primitive permutation group of degree , with primes, where . Then, one of the following holds. (a) is doubly transitive and either(i) is affine type, , and is transitive on the nonzero vectors of , or(ii) is almost simple type and is one of the following groups: , , and , where .(b) is simply primitive, and one of the following is true:(i) is affine type, , and is irreducible but not transitive on the nonzero vectors of ;(ii) is almost simple type, and , with ;(iii) is product action type, and , where , , and is a transitive subgroup of ; lies in the following list: ; with ; , with , ; with , ; with .
Remark 2. The subgroups of which are transitive on the nonzero vectors of are determined by Hering in .
2. Primitive Groups of Degree
Let be a nonempty set. Recall that the symmetric group is defined to be the group of all permutations of . A permutation group on is simply a subgroup of the symmetric group , and the size is called the degree of . If , then 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 or if .
Let . For , we use to denote the point stabilizer of with respect to . The orbit of containing is defined to be the set . It is well known that . The group is called transitive if, for any two points , there exists an element such that , which is equivalent to saying that has only one orbit on . A group is called semiregular if for every , 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 , let be a set of subsets of . Then, is said to be a partition of if is the disjoint union of ; that is, , and for . There are two trivial partitions of : and with . Let be a transitive permutation group on . A partition of is said to be -invariant, if, for any and , . In this case, is also said to be an imprimitive partition of on .
Let be transitive. Then, is called primitive if it has no nontrivial -invariant partition. If is an imprimitive partition of on , then divides , and so each transitive permutation group of prime degree is primitive. Let be the set of all -tuples of points in ; that is, Then, one can define an action of on by A group is said to be -transitive if is transitive on . A -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 , where , 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 ). 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 be a primitive permutation group, and let 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. HA (holomorph affine): is a regular elementary abelian -group for some prime , , and , where is irreducible on . AS (almost simple): is a nonabelian simple group, and . SD (simple diagonal): , where , , and . PA (product action): , where , , , and let be a primitive group of with ; then, is isomorphic to a subgroup of , with the product action, and . TW (twisted wreath product): , where , and is regular on , with .
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 , where () for some prime . Then, the type of is HA, AS, or PA.
Proof. Let be a primitive permutation group acting on a set , let be the socle of , and let . 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, . 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 , then the type of is HA or AS.
In , Guralnick classified finite simple group with a subgroup index a prime power.
Theorem 6 (Guralnick, ). Let be a nonabelian simple group, let , and let , where and is a prime. Then, one of the following holds. (a) and , with .(b), is the stabilizer of a line or a hyperplane, and .(c) and .(d) and or and .(e) and .
As a result, we have the following.
Corollary 7. Let be a nonabelian simple group acting transitively on a set with . Then, either acts doubly transitive on or acts simply primitively on , with .
Proof. Let act transitively on , where . Then, . 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 unless , which has a maximal subgroup , of index 27, and is simply primitive on . Therefore, the result is true.
We point out that if , then either or , such that is a Mersenne prime.
In , 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 . In this case, , and is an irreducible subgroup of which is transitive on nonzero vectors of . Such groups have been determined by Huppert in  for soluble case and by Hering  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 which are not transitive on the nonzero vectors of .
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 () are as follows.
Theorem 9. Let be a doubly transitive group of degree () with nonabelian socle. Then, either or or , where .
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.(1) is primitive of holomorph affine type, and .(2) is primitive of almost simple type, and one of the following holds:(i) or ;(ii), where ;(iii) or , ;(iv) or and or .
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 and , respectively. Then, the wreath product is primitive in the product action on if and only if(i) acts primitively but not regularly on ;(ii) is finite and acts transitively on .
It follows from the above result that, for , with prime, we have the next corollary.
Corollary 12. The wreath product is primitive in the product action on with , if and only if one of the following holds. (a) is a prime, is one as listed in of Theorem 10, and .(b), , is a nonregular primitive group of degree , and is a transitive subgroup of .
2.1. Proof of Theorem 1
In this section, we deal with the case of and present a proof of Theorem 1.
Lemma 13. Let be a primitive permutation group acting on a set , where . Assume that . Then, is one of the following: , where , or ; ; ; ; and .
Proof. It is clear that there are no simply primitive groups of degree 8, so is doubly transitive on . If is almost simple, then . Applying Theorem 6, we find that or ; thus, , , , or . If is affine, then by Hering’s result (see ), we have which has a cyclic normal subgroup and which is irreducible on or . Since and , , where , or .
Now, we prove Theorem 1.
Proof. Let be a primitive permutation group acting on a set , where for some prime , so is either doubly primitive or simply primitive on . By Lemma 5, we know that is of type HA, AS, or PA. Let be the socle of .
If is doubly primitive on , then by Theorem 8, the type of is HA or AS, or where is nonabelian simple. In the former case, and is transitive on the nonzero vectors of , such subgroups are determined by Hering in . In the latter case, by Theorem 9, either or or , and or where . Thus, (a) of Theorem 1 holds.
Assume now that is simply primitive on . If the type of is HA, then ; is irreducible but not transitive on the nonzero vectors of ; we obtain (i) (b) of Theorem 1. If the type of is AS, then by Corollary 7, acts simply transitively on where . Since and , or ; we have (ii) (b) of Theorem 1. Assume that the type of is PA; then, , where , , and is primitive of degree of the type AS or SD. Applying Lemma 5, we obtain that the type of is AS, so is nonabelian, and thus , with , , with , , with , , or with , which are determined by Theorem 6. Since the wreath product is of type PA, then by Corollary 12, is a transitive subgroup of . Thus, we obtain (iii) (b) of Theorem 1.
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.
Conflict of Interests
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).
W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge, UK, 1911.