# A Note on Primitive Permutation Groups of Prime Power Degree

Authors: Qian Cai; Hua Zhang
Journal: Journal of Discrete Mathematics
 (2015)

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## Abstract

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.


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## Body

## 1. 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 groupsPSL2(p) have permutation representations of degree p with p=5,7, and 11. In 1861, Mathieu discovered his famous multiply transitive groups, including two sporadic simple groups M11 and M23, 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 p-group with p 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 n, where n=2·3r, 5·3r, or 10·3r, with r 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 becamea theorem of O’Nan and Scott and finallythe 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 p2, with p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Zp≀Zp. Hence, it is a natural next step to carry out a study on transitive groups of degree p3 or, more generally, of degree pk, with p 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 degreepr, with p a prime. In the spirit of the Burnside Theorem and Dobson’s work, here we may assume that r≥3. 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.
LetG be a primitive permutation group of degree pr, with p primes, where r≥3. Then, one of the following holds. (a)
G is doubly transitive and either
(i)
G is affine type, G=Zpr⋊Gα≤AΓLr(p), and Gα is transitive on the nonzero vectors of Zpr, or(ii)
G is almost simple type and G is one of the following groups: Apr, Spr, and PSLn(q)≤G≤PΓLn(q), where qn-1/(q-1)=pr.(b)
G is simply primitive, and one of the following is true:
(i)
G is affine type, G=Zpr:Gα≤AΓLr(p), and Gα is irreducible but not transitive on the nonzero vectors of Zpr;(ii)
G is almost simple type, and PSU42≅PSp43≤G≤PSU42.2≅PSp43.2, with p=r=3;(iii)
G is product action type, and Td=soc(G)⊲G≤U≀K, where 1<d≤r, d∣r, and K is a transitive subgroup of Sd; soc(U)=T≤Spr/d lies in the following list: Apr/d; PSLn(q) with (qn-1)/(q-1)=pr/d; M11, PSL2(11) with p=11, r=d; M23 with p=23, r=d; PSU4(2) with 1<d=r/3.Remark 2.
The subgroups ofGLn(q) which are transitive on the nonzero vectors of Zqn are determined by Hering in [10].


## 2. Primitive Groups of Degreepr

LetΩ be a nonempty set. Recall that thesymmetric groupSym(Ω) is defined to be the group of all permutations of Ω. Apermutation groupG on Ω is simply a subgroup of the symmetric group Sym(Ω), and the size |Ω| is called thedegree of G. If |Ω|=n, then Sym(Ω) is also denoted by Sn. The set of the even permutations of Sn forms a subgroup of Sn, which is called thealternating group of degree n and is denoted by Alt(Ω) or An if |Ω|=n.LetG≤Sym(Ω). For α∈Ω, we use Gα to denote the point stabilizer of G with respect to α. Theorbit of G containing α is defined to be the set αG:={αg∣g∈G}. It is well known that |αG|=|G|/|Gα|. The group G is calledtransitive if, for any two points α,β∈Ω, there exists an element g∈G such that αg=β, which is equivalent to saying that G has only one orbit on Ω. A group G≤Sym(Ω) is calledsemiregular if Gα=1 for every α∈Ω, and G is calledregular if G is transitive and semiregular. Observe that any proper subgroup of a regular group is semiregular but not regular.For a setΩ, let B={B1,B2,…,Bm} be a set of subsets of Ω. Then, B is said to be a partition of Ω if Ω is the disjoint union of Bi; that is, Ω=B1∪B2⋯∪Bm, and Bi∩Bj=∅ for i≠j. There are twotrivial partitions of Ω: B={Ω} and B={B1,B2,…,Bm} with |Bi|=1. Let G be a transitive permutation group on Ω. A partition B of Ω is said to be G-invariant, if, for any g∈G and B∈B, Bg∈B. In this case, B is also said to be animprimitive partition of G on Ω.LetG≤Sym(Ω) be transitive. Then, G is calledprimitive if it has no nontrivial G-invariant partition. If B is an imprimitive partition of G on Ω, then |B| divides |Ω|, and so each transitive permutation group of prime degree is primitive. Let Ω(k) be the set of all k-tuples of points in Ω; that is,
(1)Ω(k)∶=α1,α2,…,αk∣αi∈Ω.
Then, one can define an action of G on Ω(k) by
(2)α1,α2,…,αkg=α1g,α2g,…,αkg.
A group G≤Sym(Ω) is said to be k-transitive if G is transitive on Ω(k). A 2-transitive group is also calleddoubly transitive. It is well-known that a doubly transitive group is primitive. If G is primitive but not doubly transitive, then we say that G issimply 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 groupG≤Sym(Ω), where |Ω|≥2, is primitive if and only if each of its point stabilizers is a maximal subgroup of G.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 G≤Sym(Ω) be a primitive permutation group, and let H=soc(G) be thesocle of G, that is, the product of all minimal normal subgroups of G. Then, we can give a brief description of the five types of finite primitive groups as follows.HA (holomorph affine):H is a regular elementary abelian p-group Zpm for some prime p, |Ω|=pm, and G=Zpm:Gα≤AΓLm(p), where Gα is irreducible on Zpm.AS (almost simple):H=T is a nonabelian simple group, and T⊴G≤Aut(T).SD (simple diagonal):H≅Tm, where m≥2, |Ω|=|T|m-1, and Hα≅T.PA (product action):H≅Tm, where m≥2, d<m, d∣m, and let U be a primitive group of l with soc(U)≅Td; then, G is isomorphic to a subgroup of U≀Sym(m/d), with the product action, and |Ω|=lm/d.TW (twisted wreath product):H≅Tm, where m≥6, and H is regular on Ω, with |Ω|=|T|m.Remark 4.
In the above description,T is a nonabelian simple group and U is primitive of type AS or SD.Lemma 5.
LetG be a primitive permutation group acting on a set Ω, where |Ω|=pn (n∈N) for some prime p. Then, the type of G is HA, AS, or PA.Proof.
LetG be a primitive permutation group acting on a set Ω, let H be the socle of G, and let α∈Ω. Then, by the O’Nan-Scott Theorem, the type of G is HA, AS, SD, PA, or TW. Suppose that the type of G is SD. Then, m≥2,|Ω|=|T|m-1=pa. Since T is a nonabelian simple, this is not possible. Similarly the type of G is not TW. Thus, the type of G is HA, AS, or PA.It is clear that ifn=1, then the type of G is HA or AS.In [3], Guralnick classified finite simple group with a subgroup index a prime power.Theorem 6 (Guralnick, [3]).
LetG be a nonabelian simple group, let H<G, and let |[G:H]|=pa, where a∈N and p is a prime. Then, one of the following holds. (a)
G=An and H≅An-1, with n=pa.(b)
G=PSLn(q), H is the stabilizer of a line or a hyperplane, and |[G:H]|=(qn-1)/(q-1)=pa.(c)
G=PSL2(11) and H≅A5.(d)
G=M23 and H≅M22 or G=M11 and H≅M10.(e)
G=PSU4(2)≅PSp4(3) and H≅Z24⋊A5.As a result, we have the following.Corollary 7.
LetG be a nonabelian simple group acting transitively on a set Ω with |Ω|=pa. Then, either G acts doubly transitive on Ω or G≅PSU4(2) acts simply primitively on Ω, with |Ω|=27.Proof.
LetG act transitively on Ω, where |Ω|=pa. Then, [G:Gα]=pa. Therefore, the pair (G,Gα)=(G,H) 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 G acts doubly transitive on Ω unless G≅PSU4(2), which has a maximal subgroup H≅Z24:A5, of index 27, and G is simply primitive on Ω=[G:H]. Therefore, the result is true.We point out that ifp=2, then either G=A2a or G=PSL2(q), such that q=2a-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 abelianp-group or a nonabelian simple group.It follows from the above result that a doubly transitive group is of type HA or AS.LetG be a doubly transitive group of degree pn, with p a prime. If the type of G is HA, then the socle of G is abelian, which is isomorphic to an elementary abelian group Zpn. In this case, G=Zpn:Gα≤AΓLn(p), and Gα is an irreducible subgroup of ΓLn(p) which is transitive on nonzero vectors of Zpn. Such groups G have been determined by Huppert in [13] for soluble case and by Hering [10] for insoluble case.IfG 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 ofGLn(p) which are not transitive on the nonzero vectors of Zpn.If the type ofG is AS, then G 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 pn (n≥2) are as follows.Theorem 9.
LetG be a doubly transitive group of degree pn (n≥2) with nonabelian socle. Then, either G=Apn or G=Spn or PSLk(q)≤G≤PΓLk(q), where (qk-1)/(q-1)=pn.Transitive groups of prime degree are known for a very long time and are given as follows.Theorem 10 (see [11, page 99]).
LetG be a transitive group of prime p degree. Then, one of the following statements holds.(1)
G is primitive of holomorph affine type, and Zp≤G≤AGL1(p).(2)
G is primitive of almost simple type, and one of the following holds:
(i)
G=Ap or Sp;(ii)
PSLn(q)≤G≤PΓLn(q), where (qn-1)/(q-1)=p;(iii)
G=PSL2(11) or PΓL2(11), p=11;(iv)
G=M11 or M23 and p=11 or 13.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 thatH and K are nontrivial permutation groups acting on the sets Γ and Δ, respectively. Then, the wreath product G=K≀H is primitive in the product action on Ω=Δ|Γ| if and only if(i)
K acts primitively but not regularly on Δ;(ii)
Γ is finite and H acts transitively on Γ.It follows from the above result that, for|Ω|=pn, with p prime, we have the next corollary.Corollary 12.
The wreath productG=K≀H is primitive in the product action on Ω with |Ω|=pn, if and only if one of the following holds. (a)
n is a prime, K is one as listed in (2) of Theorem 10, and Zn≤H≤Sn.(b)
n=mk, 1<k<n, K is a nonregular primitive group of degree pm, and H is a transitive subgroup of Sk.


### 2.1. Proof of Theorem1

In this section, we deal with the case ofpr=8 and present a proof of Theorem 1.Lemma 13.
LetG be a primitive permutation group acting on a set Ω, where |Ω|=8. Assume that α∈Ω. Then, G is one of the following:Z23⋊Gα, where Gα≅Z7, ΓL1(8) or SL3(2); A8; S8; PSL2(7); and PΓL2(7).Proof.
It is clear that there are no simply primitive groups of degree 8, soG is doubly transitive on Ω. If G is almost simple, then soc(G)⊲G≤Aut(G). Applying Theorem 6, we find that soc(G)≅A8 or PSL2(7); thus, G≅A8, S8, PSL2(7), or PΓL2(7). If G is affine, then by Hering’s result (see [10]), we have Gα≤ΓL1(8) which has a cyclic normal subgroup and which is irreducible on Z23 or Gα⊵SL3(2). Since ΓL1(8)≅Z7:Z3 and SL3(2)≅GL3(2), G=Z23:Gα, where Gα≅Z7, ΓL1(8) or SL3(2).Now, we prove Theorem1.Proof.
LetG be a primitive permutation group acting on a set Ω, where |Ω|=pr for some prime p, so G is either doubly primitive or simply primitive on Ω. By Lemma 5, we know that G is of type HA, AS, or PA. Let H be the socle of G.
IfG is doubly primitive on Ω, then by Theorem 8, the type of G is HA or AS, H≅Zpr or T where T is nonabelian simple. In the former case, G=Zpr:Gα≤AΓLr(p) and Gα is transitive on the nonzero vectors of Zpr, such subgroups are determined by Hering in [10]. In the latter case, by Theorem 9, either G=Apr or Spr or PSLn(q)≤G≤PΓLn(q), and H=Apr or PSLn(q) where (qn-1)/(q-1)=pr. Thus, (a) of Theorem 1 holds.
Assume now thatG is simply primitive on Ω. If the type of G is HA, then G=Zpr:Gα≤AΓLr(p); Gα is irreducible but not transitive on the nonzero vectors of Zpr; we obtain (i) (b) of Theorem 1. If the type of G is AS, then by Corollary 7, H≅PSU4(2) acts simply transitively on Ω where |Ω|=pr=27. Since H⊲G≤Aut(H) and AutH=PSU42·2, G≅PSU4(2) or PSU4(2)·2; we have (ii) (b) of Theorem 1. Assume that the type of G is PA; then, H=soc(U)d⊲G≤U≀K, where 1<d≤r, d∣r, and U is primitive of degree pr/d of the type AS or SD. Applying Lemma 5, we obtain that the type of U is AS, so soc(U)=T is nonabelian, and thus soc(U)=T=Apr/d, PSLn(q) with (qn-1)/(q-1)=pr/d, M11, PSL2(11) with p=11, r=d, M23 with p=23, r=d, or PSU4(2) with 1<d=r/3, which are determined by Theorem 6. Since the wreath product U≀K is of type PA, then by Corollary 12, K is a transitive subgroup of Sd. Thus, we obtain (iii) (b) of Theorem 1.
This completes the proof of Theorem1.The result of Theorem1 gives rise to the following natural question.Question 1.
Study imprimitive groups of degreepr with p a prime.


## 2.1. Proof of Theorem1

In this section, we deal with the case ofpr=8 and present a proof of Theorem 1.Lemma 13.
LetG be a primitive permutation group acting on a set Ω, where |Ω|=8. Assume that α∈Ω. Then, G is one of the following:Z23⋊Gα, where Gα≅Z7, ΓL1(8) or SL3(2); A8; S8; PSL2(7); and PΓL2(7).Proof.
It is clear that there are no simply primitive groups of degree 8, soG is doubly transitive on Ω. If G is almost simple, then soc(G)⊲G≤Aut(G). Applying Theorem 6, we find that soc(G)≅A8 or PSL2(7); thus, G≅A8, S8, PSL2(7), or PΓL2(7). If G is affine, then by Hering’s result (see [10]), we have Gα≤ΓL1(8) which has a cyclic normal subgroup and which is irreducible on Z23 or Gα⊵SL3(2). Since ΓL1(8)≅Z7:Z3 and SL3(2)≅GL3(2), G=Z23:Gα, where Gα≅Z7, ΓL1(8) or SL3(2).Now, we prove Theorem1.Proof.
LetG be a primitive permutation group acting on a set Ω, where |Ω|=pr for some prime p, so G is either doubly primitive or simply primitive on Ω. By Lemma 5, we know that G is of type HA, AS, or PA. Let H be the socle of G.
IfG is doubly primitive on Ω, then by Theorem 8, the type of G is HA or AS, H≅Zpr or T where T is nonabelian simple. In the former case, G=Zpr:Gα≤AΓLr(p) and Gα is transitive on the nonzero vectors of Zpr, such subgroups are determined by Hering in [10]. In the latter case, by Theorem 9, either G=Apr or Spr or PSLn(q)≤G≤PΓLn(q), and H=Apr or PSLn(q) where (qn-1)/(q-1)=pr. Thus, (a) of Theorem 1 holds.
Assume now thatG is simply primitive on Ω. If the type of G is HA, then G=Zpr:Gα≤AΓLr(p); Gα is irreducible but not transitive on the nonzero vectors of Zpr; we obtain (i) (b) of Theorem 1. If the type of G is AS, then by Corollary 7, H≅PSU4(2) acts simply transitively on Ω where |Ω|=pr=27. Since H⊲G≤Aut(H) and AutH=PSU42·2, G≅PSU4(2) or PSU4(2)·2; we have (ii) (b) of Theorem 1. Assume that the type of G is PA; then, H=soc(U)d⊲G≤U≀K, where 1<d≤r, d∣r, and U is primitive of degree pr/d of the type AS or SD. Applying Lemma 5, we obtain that the type of U is AS, so soc(U)=T is nonabelian, and thus soc(U)=T=Apr/d, PSLn(q) with (qn-1)/(q-1)=pr/d, M11, PSL2(11) with p=11, r=d, M23 with p=23, r=d, or PSU4(2) with 1<d=r/3, which are determined by Theorem 6. Since the wreath product U≀K is of type PA, then by Corollary 12, K is a transitive subgroup of Sd. Thus, we obtain (iii) (b) of Theorem 1.
This completes the proof of Theorem1.The result of Theorem1 gives rise to the following natural question.Question 1.
Study imprimitive groups of degreepr with p a prime.



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