Digraph representations of 2-closed permutation groups with a normal regular cyclic subgroup

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1 Dgraph representatons of 2-closed permutaton groups wth a normal regular cyclc subgroup Jng Xu Department of Mathematcs Captal Normal Unversty Bejng , Chna xujng@cnu.edu.cn Submtted: Mar 30, 2015; Accepted: Nov 13, 2015; Publshed: Nov 27, 2015 Mathematcs Subject Classfcatons: 05C25, 20B25 Abstract In ths paper, we classfy 2-closed (n Welandt s sense) permutaton groups whch contan a normal regular cyclc subgroup and prove that for each such group G, there exsts a crculant Γ such that Aut(Γ) = G. 1 Introducton In 1969, Welandt [15] ntroduced the concept of the 2-closure of a permutaton group. Let G be a fnte permutaton group on a set Ω, the 2-closure G (2) of G on Ω s the largest subgroup of Sym(Ω) contanng G that has the same orbts as G n the nduced acton on Ω Ω, and we say G s 2-closed f G = G (2). It seems mpossble to classfy all 2-closed transtve permutaton groups. However, certan classes of 2-closed transtve groups have been determned. For example, n [16, 17] the author determned all 2-closed odd-order transtve permutaton groups of degree pq where p, q are dstnct odd prmes. In ths paper, one of our man purposes s to classfy all 2-closed permutaton groups wth a normal regular cyclc subgroup, see Theorem 1.2. Recall that a permutaton group s regular f t s transtve and the only element that fxes a pont s the dentty. And for more nformaton about the 2-closures of permutaton groups contanng a cyclc regular subgroup, see also [7]. Another research topc of ths paper s the study of the automorphsm groups of (d)graphs. The full automorphsm group of a (d)graph Γ must be 2-closed snce any permutaton of the vertex set that preserves the orbts of Aut(Γ) on ordered pars preserves adjacency. However, not every 2-closed permutaton group s the full automorphsm group Ths work was supported by NSFC (project number , ). the electronc journal of combnatorcs 22(4) (2015), #P4.31 1

2 of some (d)graph. Therefore, the concept of 2-closed groups s more general than the concept of the full automorphsm groups of (d)graphs, and the classfcaton of 2-closed groups s closely related to the study of the full automorphsm groups of the correspondng dgraphs. In ths paper, n order to determne 2-closed groups that contan a normal regular cyclc subgroup, we also study crculant dgraphs, that s Cayley dgraphs of cyclc groups. See Secton 2 for a more detaled explanaton. Furthermore, we dscuss the followng representaton problem. A dgraph Γ wth vertex set Ω s sad to represent a permutaton group G Sym(Ω) f Aut(Γ) = G. In ths case, we also say that the permutaton group G has a dgraph representaton Γ. Dgraph representaton problem: gven a 2-closed group G, s there a dgraph Γ that represents G? Suppose the dgraph Γ represents a 2-closed group G Sym(Ω). Then for any g Sym(Ω), to determne whether g les n G we only need to test f g preserves the sngle 2-relaton gven by the arc set of Γ, nstead of checkng all G-nvarant 2-relatons. We say a dgraph Γ s arc-transtve f Aut(Γ) s transtve on the arc set of Γ. Ths means, the arc set of Γ s actually a mnmal Aut(Γ)-nvarant 2-relaton. Suppose further that the 2-closed group G can be represented by an arc-transtve dgraph Γ. Then a permutaton g les n G f and only f g leaves nvarant the mnmal G-nvarant 2-relaton gven by the arc set of Γ. We wll show that there are arc-transtve dgraph representatons for most 2-closed groups that contan a normal regular cyclc subgroup, see the remark after Lemma Replacng dgraph wth graph, we obtan the graph representaton problem whch asks for an undrected graph to represent a 2-closed group. These two questons have prevously appeared n the lterature, see for example [1, 4]. Clearly, the graph verson problem s much more complcated than the dgraph one. Snce we are nterested n understandng the concept of 2-closed groups, we concentrate on the dgraph representaton problem n ths paper. A regular permutaton group s 2-closed, and n 1980, Baba [2] proved that wth fve exceptons, every fnte regular permutaton group occurs as the automorphsm group of a dgraph. Ths s the famous DRR (dgraphcal regular representatons) problem [2]. It s proved n [14] that for any prme power q, the semlnear group ΓL(1, q) can be represented by an arc-transtve crculant dgraph. Moreover, t s shown n [16, 17] that every 2- closed odd-order transtve permutaton group of degree pq has a tournament dgraph representaton. As for graphcal representaton problem, see for example [3, 6, 8, 9, 10, 13]. In ths paper, we wll prove that every 2-closed permutaton group G wth a normal regular cyclc subgroup s the full automorphsm group of a crculant dgraph. We may suppose that G = Z n G 0 actng on Z n naturally where G 0 Aut(Z n ). We frst descrbe the necessary and suffcent condton for G 0 such that G s 2-closed. For the detaled explanaton of notaton, see Secton 2 and Secton Condtons 1.1. Let n = 2 d 1 p d 2 2 p dt t, d 1 0, d 2,..., d t 1, t 1 where p 2,..., p t are dstnct odd prmes (also wrte p 1 = 2). And let Aut(Z n ) = Aut(Z 2 d 1 ) Aut(Z d p t ) = t D 1 D 2 D t, where D s the drect factor subgroup of Aut(Z n ) that fxes each component the electronc journal of combnatorcs 22(4) (2015), #P4.31 2

3 of the elements of Z n except for the -th component. So D = Aut(Zp d ) for each. In fact D nduces a fathful acton on the subgroup Z d p. Note that the nduced acton D 1 on the subgroup Z 2 d 1 s permutaton somorphc to ( 1) 5 (d 1 3), the multplcatve group of unts of the rng Z 2 d 1 actng on the addtve group Z 2 d 1, let φ : ( 1) (5) D 1 be the correspondng group somorphsm. Let G 0 Aut(Z n ). () f 2, d = 1 and p 5, then D G 0. () f 2 and d 2, then D G 0 Z p 1. () f d 1 = 3, then D 1 G 0. (v) f d 1 4, then ether D 1 G 0 2 or D 1 G 0 = 4 and D 1 G 0 φ(5 ). The man result of ths paper s the followng theorem. Theorem 1.2. Suppose G = Z n G 0 actng on Z n naturally where G 0 Aut(Z n ). Then G s 2-closed f and only f G 0 satsfes Condtons 1.1. Moreover, f G s 2-closed then G can be represented by a crculant dgraph. 2 Prelmnary results and notaton Frst we ntroduce some concepts and notaton concernng Cayley dgraphs. Gven a fnte group H, and a subset S H\{1}, the Cayley dgraph Γ = Cay(H, S) wth respect to S s defned as the drected graph wth vertex set H and arc set AΓ = {(g, sg) g H, s S}. Moreover, a Cayley dgraph of a cyclc group s called a crculant. It s easy to check that the rght regular representaton Ĥ s contaned n Aut(Γ). In fact, a dgraph s a Cayley dgraph f and only f ts automorphsm group contans a regular subgroup. Moreover let Aut(H, S) = {σ Aut(H) S σ = S}, then each element n Aut(H, S) nduces an automorphsm of the Cayley dgraph Γ = Cay(H, S). It s proved n [10] that the normalzer of Ĥ n Aut(Γ) s Ĥ Aut(H, S). We say a Cayley dgraph Γ = Cay(H, S) s normal f Ĥ s normal n Aut(Γ), that s, Aut(Γ) = Ĥ Aut(H, S), see [10, 18]. So the automorphsm group of a normal crculant must be a 2-closed group that contans a normal regular cyclc group. Conversely, we wll show that each such 2-closed group s the automorphsm group of some normal crculant. Throughout the rest of ths paper, let Z n be an abstract cyclc group of order n and let G Sym(Z n ) be a transtve permutaton group whch contans a normal regular cyclc group Ẑn where Ẑ n = {ĝ : x xg x Z n g Z n }. (1) Therefore G s a semdrect product Ẑn G 0 for some subgroup G 0 Aut(Z n ) actng naturally on Z n. Snce Ẑn = Z n, we may also wrte G = Z n G 0 drectly. Our goal s to determne all such 2-closed groups. the electronc journal of combnatorcs 22(4) (2015), #P4.31 3

4 The mal tool used n ths paper s the Kovács-L classfcaton of arc-transtve crculants [11, 12]. Praeger and the author [14] refned the Kovács-L classfcaton and obtaned the followng theorem. Theorem 2.1. [14, Theorem 1.1] Let G = Z n G 0 Z n Aut(Z n ) actng naturally on Z n. Then, up to somorphsm, there s a unque connected Z n -crculant Γ on whch G acts arc-transtvely. Moreover ether Aut(Γ) = G or one of the followng holds. (a) n = p 5 s prme, Γ = K p, and G = AGL(1, p); (b) n = bm > 4, where b 2, p dvdes m for each prme p dvdng b, Γ = Σ[K b ]; (c) n = pm, where p s prme, 5 p < n, and gcd(m, p) = 1, Γ = Σ[K p ] p.σ, G 0 = Aut(Z p ) H Aut(Z p ) Aut(Z m ), and Σ s a connected (Z m H)-arctranstve Z m -crculant. We pont out that up to somorphsm, n the above theorem Γ can be defned as Cay(Z n, z G 0 ) where z s a generator of Z n and z G 0 s the orbt of z under G 0. Moreover, f case (b) happens, then the group Z n has a subgroup Y of order b, and Γ = Cay(Z n, S) where S s a unon of Y -cosets each consstng of generators for Z. As a smple applcaton of Theorem 2.1, we determne the 2-closed transtve permutaton groups of degree p where p s a prme. Corollary 2.2. Let p be a prme. Let G Sym(Ω) be a 2-closed transtve permutaton group of degree p. Then there exsts a dgraph representng G. Moreover, G s one of the followng. 1. The symmetrc group S p (p 2) whch s 2-transtve on Ω. 2. An affne subgroup Z p Z k where p 3, 1 k < (p 1) and k (p 1). Conversely, each group of the above two types s 2-closed. Proof. Suppose G s a 2-closed transtve permutaton group of degree p. By a classcal result of Burnsde, G s ether 2-transtve or s affne. If G s 2-transtve, then G = G (2) = S p and p 2. If G s not 2-transtve, then G = Z p Z k where p 3, 1 k < (p 1) and k (p 1). For the converse, note that S p s the full automorphsm group of the complete graph K p and so S p s ndeed 2-closed. Next, let G = Z p Z k where p 3, 1 k < (p 1) and k (p 1). By Theorem 2.1, there s a connected arc-transtve crculant Γ of order p such that Aut(Γ) = G, and so G s 2-closed. Remark: If p = 2, 3 then S p = Z p Aut(Z p ) s 2-closed; and f p 5 then Z p Aut(Z p ) s not 2-closed. We also need the followng theorem. Theorem 2.3. [5, Theorem 5.1] Let G 1 Sym(Ω 1 ) and G 2 Sym(Ω 2 ) be transtve permutaton groups. Consder the natural product acton of G 1 G 2 on Ω 1 Ω 2. Then (G 1 G 2 ) (2) = G (2) 1 G (2) 2. the electronc journal of combnatorcs 22(4) (2015), #P4.31 4

5 Fnally, we fx the followng notaton. Let A Sym(Ω). Suppose that A B s the setwse stablzer of B Ω and g A B, we denote A B B to be the nduced permutaton group on B by A B and denote g B to be the nduced permutaton on B by g. 3 2-closed groups contanng a normal regular cyclc group In ths secton we classfy 2-closed groups G that contan a normal regular cyclc group Z n. Wth notaton n Secton 2, we may suppose that G = Z n G 0 Z n Aut(Z n ) actng naturally on Z n. We frst handle the specal case that n s a prme power n Subsecton 3.1 and Subsecton 3.2. The notaton needed for the statement of Theorem 1.2 s gven n Subsecton and the proof s gven n Subsecton The case n = p d wth p an odd prme Let n = p d where p s an odd prme and d 2 s an nteger. Then Aut(Z n ) = Z (p 1) Z p d 1 s a cyclc group. We take α Aut(Z n ) such that o(α) = p, then there exsts γ Aut(Z n ) wth order p d 1 such that α = γ pd 2. We frst look at the acton of α on Z n. Let H = Z p d 1 be the unque subgroup of Z n of order p d 1. Let N = Z n Aut(Z n ). Then the cosets of H form a block system B of N on Z n. Denote B = {B 1 = H, B 2,..., B p }. Snce the elements n B 2,..., B p are of order p d, γ fxes each block setwse and γ B s a p d 1 cycle for each 2. However, γ fxes the pont 1 H = B 1, so the order of γ B 1 s strctly less than p d 1. It then follows that α fxes B 1 pontwse and s fxed pont free on each B for 2. On the other hand, let N B B be the nduced permutaton group of the setwse stablzer N B on B. Then N B B = Ẑp K d 1 and K = Aut(Zp d 1), (Ẑpd 1 s defned n equaton (1)). For each 2, snce γ B s fxed pont free, we have that γ B = ŷ B τ where 1 y H Z n and τ K. Snce τ normalzes Ẑp d 1, (γb ) 2 = ŷ B (τŷ B τ 1 )ττ = a 2 τ 2 where a 2 s some element n Ẑp d 1. By nducton, we have that for each k 1, (γb ) k = a k τ k where a k s some element n Ẑp d 1. Snce γb s of order p d 1 and Ẑp K d 1 = {1}, we have that τ pd 1 = 1. Snce τ Aut(Z p d 1) = Z p 1 Z p d 2, τ pd 2 = 1. Recall that α = γ pd 2, t then follows that α B s ˆx B for some x Z n wth order p. Note that x may not equal x j for 2 < j p, but they are all of order p. We have proved the followng lemma. Lemma 3.1. Let α Aut(Z p d) wth order p. Let B = {B 1 = H, B 2,..., B p } be the cosets of the subgroup H where H < Z p d s of order p d 1. Then α fxes B 1 = H pontwse and for each 2, α B s ˆx B for some x Z n wth order p. Corollary 3.2. Let n = p d and Z n = z. Let Z p Z n be the subgroup of order p. Suppose that G = Z n G 0 where G 0 Aut(Z n ). Then the coset zz p z G 0 f and only f p G 0. Remark: Let S = z G 0 and Γ = Cay(Z n, S). If case (b) of Theorem 2.1 occurs for Γ, then zz p z G 0. That s why we consder ths corollary. the electronc journal of combnatorcs 22(4) (2015), #P4.31 5

6 Proof. Let Aut(Z p d) = µ γ = Z p 1 Z p d 1 and α = γ pd 2. Then p G 0 f and only f α G 0. Let B = {B 1 = H, B 2,..., B p } be the cosets of the subgroup H where H < Z p d s of order p d 1. Then t s easy to show that µ fxes B 1 setwse, and permutes B 2,..., B p as a (p 1)-cycle. By Lemma 3.1, f α G 0 then zz p z G 0. Conversely, suppose that zz p z G 0. Note that the generator z B k for some k 2 and zz p B k. By the acton of µ and γ, we conclude that α G 0. Proposton 3.3. Let n = p d where p s an odd prme and d 2. Let G = Z n G 0 Z n Aut(Z n ) actng naturally on Z n. Then G s 2-closed f and only f G 0 Z p 1. Moreover, f G s 2-closed then G can be represented by an arc-transtve crculant. Proof. As defned at the begnnng of Subsecton 3.1, let α Aut(Z p d) be an element of order p. Let B = {B 1 = H, B 2,..., B p } be the cosets of the subgroup H where H < Z p d s of order p d 1. Suppose frst that G 0 Z p 1, that s p G 0, then α G 0. By Lemma 3.1, α fxes B 1 = H pontwse and for each 2, α B s ˆx B for some x Z n wth order p. Let 1 β Sym(Z n ) such that β fxes every element of B 1,..., B p 1 and β Bp = α Bp. That means β Bp = ˆx Bp p, (recall that ˆx : z zx for any z Z n ). We clam that β (Z p d α ) (2) and so β G (2). Take any par (y 1, y 2 ) Z n Z n. If both y 1 and y 2 belong to B p, then (y 1, y 2 ) β = (y 1 x p, y 2 x p ) s n the orbtal (y 1, y 2 ) G. Suppose next that exactly one of {y 1, y 2 } les n B p, say y 2 B p. Snce the stablzer G y1 s the conjugate of G 0 n G by an element n Ẑn, a conjugate of α, say ρ, s n G y1. Therefore β Bp equals (ρ j ) Bp for some j {1,..., p 1}, and so (y 1, y 2 ) β (y 1, y 2 ) G. It then follows that β (Z p d α ) (2) G (2). However, snce β fxes B 1 and B 2 pontwse, β / Z p d Aut(Z p d), and so β / G and G s not 2-closed. Suppose next that G 0 Z p 1. Let S = z G 0 where z Z p d s an element of order p d and let Γ = Cay(Z n, S). Snce (p, G 0 ) = 1, p S and so S s not a unon of cosets of any subgroup of Z n. By Theorem 2.1, Aut(Γ) = G and so G s 2-closed. Ths completes the proof. Remark: In above proof, note that β s n (Z p d α ) (2). Hence we actually proved that (Z p d α ) (2) Z p d Aut(Z p d) where α Aut(Z p d) s of order p. 3.2 The case n = 2 d for d 2 Notaton: For convenence, n ths subsecton we wrte Z n addtvely as the group Z n of ntegers modulo n, so n ths case Ẑ n = Ẑn = {ˆx : g g + x x Z n }. Moreover Aut(Z n ) s the multplcatve group Z n so that Aut(Z n ) denotes the map j j. the electronc journal of combnatorcs 22(4) (2015), #P4.31 6

7 3.2.1 d = 2: In ths case, Aut(Z 4 ) = ( 1) = Z 2. We have the followng result. Lemma 3.4. Suppose that Ẑ4 G Ẑ4 ( 1) = D 8. Then G s 2-closed and s the full automorphsm group of an arc-transtve crculant. Proof. Ether G = Z 4 s regular or G = D 8. Note that Aut(Cay(Z 4, {1})) = Z 4 and Aut(Cay(Z 4, {1, 1})) = D 8 = Z 4 Z 2, ths proves the lemma. Remark: By [14, Lemma 2.3], a connected arc-transtve crculant Γ s both normal and of lexcographc product form f and only f Γ = Cay(Z 4, {1, 1}) and Aut(Γ) = Z 4 Aut(Z 4 ). In ths case the orbt 1 Aut(Z 4) = {1, 3} = 1 + Z 2 s a coset of Z d 3: In ths case, Aut(Z n ) = ( 1) 5 = Z 2 Z 2 d 2. Denote N = Ẑn Z n. Let H be the unque subgroup of Z n wth order 2 d 2. Let B 0 = H, B 1 = 1 + H, B 2 = 2 + H, B 3 = 3 + H be the cosets of H, then B = {B 0, B 1, B 2, B 3 } forms a complete block system of N on Z n. We frst study the acton of 5. By computaton 5 preserves each block B, we determne the nduced permutaton (5 ) B next. Snce B 1 B 3 conssts of all elements of order 2 d, (5 ) B 1 and (5 ) B 3 are 2 d 2 -cycles. As B 0 = 4 = Z 2 d 2 and B 0 B 2 = 2 = Z 2 d 1, t s easy to deduce that (5 ) B 2 s a product of two 2 d 3 -cycles (f d = 3, then (5 ) B 2 s trval). Therefore the orders of (5 ) B 1 and (5 ) B 3 are 2 d 2, the order of (5 ) B 2 s 2 d 3, and the order of (5 ) B 0 s 2 d 4 (f d = 3, then the order s 1). Case 1: d = 3 In ths case, n = 8 and Aut(Z 8 ) = ( 1) 5 = Z 2 Z 2. By computaton, 5 fxes B 0 and B 2 pontwse, and the nduced acton (5 ) B 1 = ˆ4 B 1 and (5 ) B 3 = ˆ4 B 3. The element ( 1) fxes B 0 pontwse and (( 1) ) B 2 = ˆ4 B 2. Lemma 3.5. Let Z 8 = z. Suppose that G = Z 8 G 0 where G 0 Aut(Z 8 ) = ( 1) 5. Then the coset z + Z 2 z G 0 f and only f 5 G 0 where Z 2 = 4 s the subgroup of order 2. Proof. Note that both z and z + Z 2 are contaned n B 1 or B 3 and ( 1) nterchanges two blocks B 1 and B 3. The result follows from the analyss of the actons of ( 1) and 5 easly. Proposton 3.6. Wth above notaton, let G = Z 8 G 0 where G 0 Aut(Z 8 ) = ( 1) 5. Then 1. f G 0 = Aut(Z 8 ) then G s not 2-closed. 2. f G 0 Aut(Z 8 ) and G 0 5, then G s 2-closed and can be represented by an arc-transtve crculant. 3. f G 0 = 5, then G s 2-closed and can be represented by a crculant. the electronc journal of combnatorcs 22(4) (2015), #P4.31 7

8 Proof. (1) Suppose frst that G 0 = Aut(Z 8 ). Let β S 8 such that β fxes B 0, B 1 and B 3 pontwse and β B 2 = ˆ4 B 2. Take any par (y 1, y 2 ) Z 8 Z 8. If both y 1 and y 2 belong to B 2, then (y 1, y 2 ) β = (y 1, y 2 )ˆ4 s n the orbtal (y 1, y 2 ) G. Suppose next that exactly one of {y 1, y 2 } belongs to B 2, say y 2 B 2. It s straghtforward to check that (y 1, y 2 ) β = (y 1, y 2 ) ( 1) f y 1 B 0. Let G 1 be the pont stablzer of pont 1, then G 1 s the conjugate of G 0 by ˆ1 Ẑn. Let α 1 be the correspondng conjugate of 5 n G 1. It follows that (y 1, y 2 ) β = (y 1, y 2 ) α 1 f y 1 B 1 B 3. Hence β G (2). However snce β fxes 0 and 1, β / G and so G s not 2-closed. (2) In ths case, 5 / G 0. Let S = 1 G 0 and let Γ = Cay(Z 8, S). It follows from Lemma 3.5 and Theorem 2.1 that G = Aut(Γ) and s 2-closed. (3) Fnally we show that Z 8 5 s 2-closed. Let S 1 = 1 5 = {1, 5} and S 2 = 2 5 = {2}. Let Γ = Cay(Z 8, S 1 S 2 ). By [12, Theorem 1.3], t s easy to deduce that Γ s not arc-transtve. Suppose g Aut(Γ) such that g fxes 0 and 1, t s straghtforward to check that g = 1. We conclude that Aut(Γ) = Z 8 5 as requred. Case 2: d 4 Let α = (5 ) 2d 4 be an element of order 4 n 5. By the analyss of acton of 5, we deduce that α fxes B 0 pontwse and o(α B 2 ) = 2, o(α B 1 ) = o(α B 3 ) = 4. Suppose frst that d = 4, then α = 5. By drect computaton, α B 2 = ˆ8 B 2, α B 1 = ˆ4 B 1 and α B 3 = 4 B 3. Next suppose d 5. Denote N = Ẑn Z n. Note that N B B = Ẑ 2 d 2 K where K = Aut(Z2 d 2) for each {1, 2, 3}. Snce (5 ) B s fxed pont free on B for = 1, 2, 3, (5 ) B = ŷ B τ where 0 y Z n and τ K. Snce τ normalzes Ẑ2 d 2, ((5 ) B ) 2 = ŷ B (τ ŷ B τ 1 )τ τ = a 2 τ 2 where a 2 s some element n Ẑ2d 2. By nducton, we have that for each k 1, ((5 ) B ) k = a k τ k where a k s some element n Ẑ2 d 2. Snce τ Aut(Z 2 d 2) and d 5, τ 2d 4 = 1. By the order of α B, we have that α B = ˆx B, where x 1, x 3 Z n are of order 4 and x 2 = 2 d 1 s the unque nvoluton n Z n. In addton, 2x 1 = 2x 3 = 2 d 1. Therefore we have proved the followng lemma. Lemma 3.7. Suppose d 4. Wth above notaton, let α = (5 ) 2d 4 be an element of order 4 n 5. Then α fxes B 0 pontwse, α B 2 = ( 2 d 1 ) B 2, α B 1 = ˆx B 1 1 for some x 1 Z n wth order 4 and α B 3 = ˆx B 3 3 for some x 3 Z n wth order 4. Corollary 3.8. Let n = 2 d for d 4 and let Z n = z. Suppose that G = Z n G 0 where G 0 Aut(Z n ) = ( 1) 5. Let α 5 be of order 4. Then 1. the coset z + Z 4 z G 0 f and only f α G 0 where Z 4 Z n s the subgroup of order the coset z + Z 2 z G 0 f and only f α 2 G 0 where Z 2 Z n s the subgroup of order 2. Proof. By Lemma 3.7, we have that z + Z 4 z G 0 f α G 0 and z + Z 2 z G 0 f α 2 G 0. Wth the notaton n Lemma 3.7, suppose that z + Z 4 z G 0. Note that z B 1 or B 3 and z + Z 4 B 1 or B 3 respectvely. Snce ( 1) nterchanges B 1 and B 3, t s easy to deduce that α G 0. Smlarly, f z + Z 2 z G 0 then α 2 G 0. the electronc journal of combnatorcs 22(4) (2015), #P4.31 8

9 Proposton 3.9. Wth above notaton, let G = Z n G 0 Z n Aut(Z n ) where n = 2 d for d 4. If α = (5 ) 2d 4 G 0, then (Z n α ) (2) Z n Aut(Z n ). In partcular, G s not 2-closed on Z n. Proof. Let 1 β Sym(Z 2 d) such that β fxes B 0, B 2, B 3 pontwse and β B 1 = (2 d 1 ) B 1 s of order 2. Therefore β B 1 = (α 2 ) B 1. We wll show next that β (Z 2 d α ) (2) G (2). Take any par (y 1, y 2 ) Z n Z n. If both y 1 and y 2 belong to B 1, then (y 1, y 2 ) β = 2 (y 1, y 2 ) d 1 s n the orbtal (y 1, y 2 ) G. Suppose next that exactly one of {y 1, y 2 } belongs to B 1, say y 2 B 1. By Lemma 3.7, (y 1, y 2 ) β = (y 1, y 2 ) α2 f y 1 B 0 or B 2. Let G 3 be the pont stablzer of pont 3, then G 3 s the conjugate of G 0 by ˆ3 Ẑn. Let α 3 be the correspondng conjugate of α n G 3, t follows from Lemma 3.7 that (y 1, y 2 ) β = (y 1, y 2 ) α 3 f y 1 B 3. Thus β (Z 2 d α ) (2) G (2). However snce β fxes B 0 and B 3 pontwse, β / Z 2 d Aut(Z 2 d) and so (Z 2 d α ) (2) Z 2 d Aut(Z 2 d). In partcular G s not 2-closed. Next we wll show that f α / G 0 then G s 2-closed. Note that α / G 0 s equvalent to the condton that ether G 0 2 or G 0 = 4 and G 0 5. We frst dscuss the case that α 2 / G 0. Lemma Wth above notaton, let n = 2 d for d 4. Let G = Z n G 0. Suppose α 2 / G 0. Then G s the full automorphsm group of an arc-transtve crculant and so G s 2-closed. Proof. Let S = 1 G 0 be the orbt of 1 under G 0, and let Γ = Cay(Z n, S). Snce α 2 / G 0, t follows from corollary 3.8 that S s not a unon of cosets of any subgroup of Z n. By Theorem 2.1, Aut(Γ) = G as requred. It remans to show that f G = Z n G 0 where α 2 G 0 but α / G 0 then G s the full automorphsm group of some crculant. We wll prove ths n Proposton 3.15 when we handle the more general case. 3.3 The general case The notaton for the man theorem. We explan Condtons 1.1 n more detal frst. Let n = 2 d 1 p d 2 2 p dt t, d 1 0, d 2,..., d t 1, t 1 where p 2,..., p t are dstnct odd prmes. For convenence, we also wrte p 1 = 2. In addton, the noton p d n means pd n but pd +1 n. Let G = Ẑn G 0 actng on Z n naturally where G 0 Aut(Z n ). In order to reduce the proof n the general case to the prme power case, we choose the product acton form to descrbe G. Let Z m be the unque subgroup of Z n of order m for m n. Then we may wrte Z n = Z 2 d 1 Z p d 2 2 Z p d t t = {(z 1,..., z t ) = z 1 z 2 z t z Z d p, where p 1 = 2}. the electronc journal of combnatorcs 22(4) (2015), #P4.31 9

10 For any g = (g 1,..., g t ) Z n, we have ĝ : (z 1,..., z t ) (z 1 g 1,..., z t g t ). Moreover, Aut(Z n ) = Aut(Z 2 d 1 ) Aut(Z d p t ) = D 1 D 2 D t, t where D s the drect factor subgroup of Aut(Z n ) that fxes each component of the elements of Z n except for the -th component. So D = Aut(Zp d ). In fact D nduces a fathful acton on the subgroup Z d p. Wth notaton n 3.2, f d 1 3 then the nduced acton D 1 on the subgroup Z 2 d 1 s permutaton somorphc to ( 1) 5 (d 1 3), the multplcatve group of unts of the rng Z 2 d 1 actng on the addtve group Z 2 d 1. Let φ : ( 1) (5) D 1 be the correspondng group somorphsm. The normalzer of Ẑn n Sym(Z n ) s N = Ẑn Aut(Z n ) = (Ẑ2 d 1 Aut(Z 2 d 1 )) (Ẑp d t t Aut(Z d p t )) t actng on Z n by the natural product acton. Therefore G = Ẑn G 0 N has the natural product acton. We need the followng two easy observatons n the proof below. (1) Note that when 2, Aut(Z p d α / G 0 where α D = Zp 1 Z p d 1 ) = Z p 1 Z d p 1 s of order p.. Condtons 1.1 [] s equvalent to (2) When = 1 and d 1 4, denote α 1 = φ((5 ) 2d 1 4 ) D 1, then the order of α 1 s 4. Condtons 1.1 [v] s equvalent to α 1 / G The proof of Theorem 1.2. Lemma Wth notaton n Subsecton 3.3.1, suppose G = Ẑn G 0 where G 0 Aut(Z n ). If G 0 fals to satsfy one of condtons 1.1, then G s not 2-closed. Proof. If condton () does not hold, then there exsts an odd prme p 5 where 2 such that p n and D G 0. In ths case we take K = Ẑp D. By hypothess, K s the subgroup of G whch fxes each component of elements of Z n except for the -th component. Hence the acton of K on Z n s the product acton of K {1} on Z n = Z p Z n p K = K acts on Z p where naturally. It follows from Theorem 2.3 that K (2) = (K) (2) {1}. By the remark after Corollary 2.2, (K) (2) Z p Aut(Z p ). Snce G (2) K (2), we have that G s not 2-closed n ths case. If condton () does not hold, then there exsts an odd prme p where 2 such that p d n and d 2. Snce α G 0 n ths case, we take K = Ẑp d acton of K on Z n s the product acton of K {1} on Zn = Z p d α G. Hence the Z n p d acts on Z d p naturally. By the remark after Proposton 3.3, (K) (2) Z d p The same argument as above proves that G s not 2-closed n ths case ether. where K = K Aut(Z d p ). Suppose 2 d 1 n and d 1 3, suppose also that ether condton () or (v) fals. Take K = Ẑ8 D 1 f d 1 = 3 and take K = Ẑ2 d 1 α 1 f d 1 4. By the same argument as above, t follows from Proposton 3.6(1) and Proposton 3.9 that G s not 2-closed. the electronc journal of combnatorcs 22(4) (2015), #P

11 Lemma Wth notaton n Subsecton 3.3.1, suppose G = Ẑn G 0 where G 0 Aut(Z n ) and G 0 satsfes Condtons 1.1. Let S = z G 0 where Z n = z, and let Γ = Cay(Z n, S). Then exactly one of the followng holds. 1. G s the full automorphsm group of Γ and so G s 2-closed and can be represented by an arc-transtve crculant d 1 n, d 1 4, and α 2 1 G 0 D n, and D 1 G 0 = φ(5 ) = Z n = 4m where m > 1 s odd. D 1 G 0 = D 1 = Z2, that s G 0 = Aut(Z 4 ) K where K Aut(Z m ). Moreover, n the latter three cases, Γ = Σ[K 2 ] s a lexcographc product and the pontwse stablzer of {1, z} n Aut(Γ) preserves each coset of Z 2. Proof. Suppose that G s not the full automorphsm group of Γ. By the condton (), for any odd prme p 5 such that p n, we have G 0 Aut(Z p ) H for some H Aut(Z n/p ). It then follows from Theorem 2.1 that case (b) of Theorem 2.1 occurs for Γ. That s n = bk > 4 where b 2 and Γ = Σ[K b ]. Moreover, the group Z n has a subgroup Y of order b and S s a unon of Y -cosets each consstng of generators for Z n. Recall that n = 2 d 1 p d 2 2 p dt t. Suppose that p j b for some j {1,..., t}. Then zz pj S where Z pj s the subgroup of order p j and d j 2 by Theorem 2.1 (b). Let z = (z 1,..., z t ) where z s a generator of Z d p for each. Thus zz pj S = z G 0 mples that z j Z pj z D j G 0 j n the j-th component. By Corollary 3.2, the condton () mples that b = 2 l s a power of 2. Smlarly, by Corollary 3.8, Lemma 3.5 and the acton of Aut(Z 4 ), the condton () and (v) mply that b must be 2 and one of cases 2-4 happens. Suppose next that one of cases 2-4 occurs. Thus Γ = Σ[K 2 ] where Σ = Cay(Z n /Z 2, S) and S = {sz 2 s S}. Moreover, by [14, Lemma 2.3], the set {xz 2 x Z n } forms a block system of Aut(Γ), and so Aut(Γ) = Z 2 Aut(Σ). Let G 0 = G 0 / α1 2 n case 2, and let G 0 = G 0 /(D 1 G 0 ) n case 3 or 4. Then G 0 Aut(Z n /Z 2 ) and S = (zz 2 ) G 0. Note that G 0 satsfes Condtons 1.1, t follows that S s not the unon of cosets of any subgroup of Z n /Z 2. By Theorem 2.1, Σ s normal and Aut(Σ) = (Z n /Z 2 ) G 0. Therefore the pontwse stablzer of {1, z} n Aut(Γ) preserves each coset of Z 2. Remark: Suppose G satsfes Condtons 1.1. By the above lemma, G can be represented by an arc-transtve crculant f and only f G does not arse n any of the cases 2-4 of Lemma Next we wll show that f one of cases 2-4 occurs then there exsts a crculant Γ whch s not arc-transtve such that Aut(Γ) = G. We dscuss case 4 frst. Lemma Suppose n = 4m where m > 1 s odd and G = Z 4m G 0 where G 0 = Aut(Z 4 ) K and K Aut(Z m ). Suppose further that G 0 satsfes Condtons 1.1. Then G s 2-closed and can be represented by a crculant. the electronc journal of combnatorcs 22(4) (2015), #P

12 Proof. Let z = z 1 z 2 Z 4m where z 1 s a generator of Z 4 and z 2 s a generator of Z m. Let S 1 = z G 0 and Γ 1 = Cay(Z 4m, S 1 ). By Lemma 3.12, S 1 s the unon of some cosets of Z 2 = z1. 2 Let S 2 = z G 0 2 Z m and Γ 2 = Cay(Z 4m, S 2 ). Thus B 0 = Z m, B 1 = zz m, B 2 = z 2 Z m and B 3 = z 3 Z m are the connected components of Γ 2. Let S = S 1 S 2 and Γ = Cay(Z 4m, S). Suppose frst that Γ s arc-transtve. Note that S 1 conssts of elements of order 4m and S 2 contans elements of order m. We observe that S s not the unon of cosets of any subgroup. By [12, Theorem 1.3], Γ = Σ[K b ] b.σ where n = br, 4 b < n and gcd(b, r) = 1. Thus wrtng Z n = Y M wth Y = Z b and M = Z r, we have that S = Y \{1} T and T M \{1}. Analyzng the orders of elements of S, we have that b = p s prme, p 5 and p m as (b, r) = 1. As z G 0 Y \ {1} T, D = Aut(Zp ) G 0, contradctng the condton (). Thus Γ s not arc-transtve. Let P be the pont stablzer of Aut(Γ) on vertex 1. Snce P G 0, P has two orbts S 1 and S 2 and so Aut(Γ) = Aut(Γ 1 ) Aut(Γ 2 ) Assume that g Aut(Γ) fxng 1 B 0 and z B 1. Consder z 2 B 2 zs 1 whch s adjacent to z. It follows from Lemma 3.12 that g fxes each coset of Z 2 = z1. 2 Hence (z 2 ) g {z 2, z 2 z1} 2 = z 2 Z 2 and g fxes both z B 1 and zz1 2 B 3. Moreover, as g Aut(Γ 2 ), we conclude that g must fx B 0, B 1, B 2 and B 3 setwse. Therefore, g fxes z 2. Contnung n ths fashon, we conclude that g fxes z 3, z 4,... and so on. Thus g = 1 and P = G 0. It follows that Aut(Γ) = G as requred. It remans to handle case 2 and case 3 n Lemma By Lemma 3.12, we may suppose that 8 n and G = Ẑn G 0 where G 0 Aut(Z n ). Let S 1 = z G 0 where Z n = z and let S 2 = (z 2 ) G 0 Z n/2 = z 2. We construct Γ = Cay(Z n, S 1 S 2 ). We wll show that Γ can represent G n both case 2 and case 3. In order for provng ths, let Γ 1 = Cay(Z n, S 1 ) and Γ 2 = Cay(Z n, S 2 ) we need to study Γ 1 and Γ 2. Note that Γ 1 has been studed n Lemma We study Γ 2 n the followng lemma. Lemma Suppose that case 2 or 3 of Lemma 3.12 occurs. Wth above notaton, we have that Γ 2 = 2.Cay( z 2, S 2 ). Let A 3 = Aut(Cay( z 2, S 2 )) and A 2 = Aut(Γ 2 ). Then A 2 = A 3 Z 2. Moreover, Cay( z 2, S 2 ) s a normal arc-transtve crculant and A 3 = z 2 G z2 0. Proof. Let 1 = z 2 and 2 = z z 2. Then Γ 2 = 2.Cay( z 2, S 2 ) such that 1 and 2 are two connected components of Γ 2. Thus A 2 = A 3 Z 2. Let G 0 = G 0 / α1 2 n case 2, and let G 0 = G 0 /(D 1 G 0 ) n case 3. Note that G 0 preserves 1, t s easy to check that the nduced permutaton group G 1 0 = G0 and G 1 0 Aut( z 2 ). Also S 2 = (z 2 ) G 1 0 s an orbt of G 1 0. Snce G 0 satsfes condtons n Theorem 1.2, S 2 s not the unon of cosets of any subgroup of z 2. By Theorem 2.1 and Condtons 1.1, we conclude that Cay( z 2, S 2 ) s normal and Aut(Cay( z 2, S 2 )) = z 2 G 1 0. Proposton Wth notaton n Subsecton 3.3.1, suppose G = Ẑn G 0 where G 0 Aut(Z n ) and G 0 satsfes Condtons 1.1. Suppose further that case 2 or 3 of Lemma 3.12 occurs. Let S 1 = z G 0 where Z n = z and let S 2 = (z 2 ) G 0. Let Γ = Cay(Z n, S 1 S 2 ) and let P be the pont stablzer of vertex 1 n Aut(Γ). Then the electronc journal of combnatorcs 22(4) (2015), #P

13 1. Γ s not arc-transtve, and S 1, S 2 are two orbts of P. 2. For any g Aut(Γ) such that g fxes 1 and z, we have that g = Aut(Γ) = G = Z n G 0. So Γ s normal and G s 2-closed. Proof. (1) Suppose, to the contrary, that Γ s arc-transtve. Note that S 1 conssts of elements of order n and S 2 contans elements of order n/2 n. Also observe that S s not the unon of cosets of any subgroup. By [12, Theorem 1.3], Γ = Σ[K b ] b.σ, where n = br, 4 b < n and gcd(b, r) = 1. Thus wrtng Z n = Y M wth Y = Z b and M = Z r, we have that S = Y \ {1} T and T M \ {1}. Analyzng the orders of elements of S, by condtons ()() we have that b = 4. As (b, r) = 1, 4 n, contradctng the fact that 8 n. Thus Γ s not arc-transtve. As P G 0, S 1, S 2 are two orbts of P. (2) Let Γ 1 = Cay(Z n, S 1 ), Γ 2 = Cay(Z n, S 2 ) and A 1 = Aut(Γ 1 ), A 2 = Aut(Γ 2 ). It follows from (1) that Aut(Γ) = A 1 A 2. Let g Aut(Γ) such that g fxes 1 and z. By Lemma 3.12, g preserves each coset of Z 2 and so (z 2 ) g {z 2, z 2 z n/2 }. Moreover, snce z 2 S 2 and g preserves S 2, we have (z 2 ) g S 2. By the proof of Lemma 3.14, we have that z 2 Z 2 S 2 and so z 2 z n/2 / S 2. Thus g fxes z 2. Let 1 = z 2 and 2 = z z 2 be two connected components of Γ 2. By Lemma 3.14, g 1 Aut( z 2 ) fxes 1 pontwse. Now g fxes z and z 2 and consder (z 3 ) g. Usng the same argument we deduce that g fxes 2 pontwse and so g = 1. (3) It follows from (2) that P = G 0 and so A = G = Ẑn G 0. Therefore Γ s normal and G s 2-closed on Z n. Theorem 1.2 now follows from Lemma 3.11, Lemma 3.12, Lemma 3.13 and Proposton References [1] B. Alspach, On constructng the graphs wth a gven permutaton group as ther group, Proc. Ffth Southeastern Conf. Combn., Graph Theory and Computng (1974), [2] L. Baba, Fnte dgraphs wth gven regular automorphsm groups, Perodca Mathematca Hungarca Vol. 11 (4), (1980), [3] L. Baba and C.D. Godsl,On the automorphsm groups of almost all Cayley graphs, Europ. J. Combn. 3 (1982) [4] L. Baba, Automorphsm groups, Isomorphsm, Reconstructon, Chapter 27 of the Handbook of Combnatorcs, , Edted by R. L. Graham, M. Grotschel and L. Lovasz, North-Holland, [5] P. J. Cameron, M. Gudc, G. A. Jones, W. M. Kantor, M. H. Kln, D. Marušč, and L. A. Nowtz, Transtve permutaton groups wthout semregular subgroups, J. London Math. Soc. 66 (2002), the electronc journal of combnatorcs 22(4) (2015), #P

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