A characterization of regular embeddings of -dimensional cubes

J Algebr Comb (2011) 33: 215–238 DOI 10.1007/s10801-010-0242-8 Classification of regular embeddings of n-dimensional cubes Domenico A. Catalano · Marston D.E. Conder · Shao Fei Du · Young Soo Kwon · Roman Nedela · Steve Wilson Received: 25 October 2009 / Accepted: 14 June 2010 / Published online: 9 July 2010 © Springer Science+Business Media, LLC 2010 Abstract An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Qn were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n = 2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Qn for all n. In particular, we show that for all even n (= 2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group Sn such that σ fixes n, preserves the set of all pairs Bi = {i, i + m} for 1 ≤ i ≤ m, and induces D.A. Catalano Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal e-mail: [email protected] M.D.E. Conder () Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e-mail: [email protected] S.F. Du School of Mathematical Sciences, Capital Normal University, Beijing 100048, China e-mail: [email protected] Y.S. Kwon Department of Mathematics, Yeungnam University, Kyongsan 712-749, Republic of Korea e-mail: [email protected] R. Nedela Mathematical Institute, Slovak Academy of Sciences, 975 49 Banská Bystrica, Slovakia e-mail: [email protected] S. Wilson Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, USA e-mail: [email protected] 216 J Algebr Comb (2011) 33: 215–238 the same permutation on this set as the permutation Bi → Bf (i) for some additive bijection f : Zm → Zm . We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n. Keywords Hypercubes · Cubes · Regular maps · Regular embeddings · Chiral 1 Introduction A (topological) map is a cellular decomposition of a closed surface. A common way to describe such a map is to view it as a 2-cell embedding of a connected graph or multigraph X into the surface S. The components of the complement S \ X are simply-connected regions called the faces of the map (or the embedding). An automorphism of a map M = (X, S) is an automorphism of the underlying (multi)graph X which extends to a self-homeomorphism of the supporting surface S. It is well known that the automorphism group of a map acts semi-regularly on the set of all incident vertex-edge-face triples (sometimes called the flags of M); in other words, every automorphism is uniquely determined by its effect on a given flag. If for any given flag (v, e, f ) the automorphism group contains two automorphisms that induce (respectively) a single cycle on the edges incident with v and a single cycle on the edges incident with f , then the map M is called rotary. In the orientable case, this condition implies that the group of all orientation-preserving automorphisms of M acts regularly on the set of all incident vertex-edge pairs (or arcs) of M, and we call M an orientably-regular map. Such maps fall into two classes: those that admit also orientation-reversing automorphisms, which are called reflexible, and those that do not, which are chiral. In the non-orientable case (and in the reflexible case), the automorphism group acts regularly on flags, while in the chiral case, there are two orbits on flags, such that the two flags associated with each arc lie in different orbits. A regular embedding (or more technically, a rotary embedding) of a graph X is then a 2-cell embedding of X as a rotary map on some closed surface. Classification of rotary maps by their underlying graphs is one of the central problems in topological graph theory. An abstract characterization of graphs having regular embeddings was given by Gardiner et al. in [12]. The classification problem has been solved only for few families of graphs, including the complete graphs [1, 13, 14], their canonical double covers [23], and complete multipartite graphs Kp,p,...,p for prime p [8, 10]. Particular contributions towards the classification of regular embeddings of complete bipartite graphs Kn,n can be found in papers [6, 7, 16, 17, 19, 20, 26], and this classification was recently completed by Gareth Jones [15]. In this paper we focus on the classification of regular embeddings of n-dimensional cubes Qn . The existence of at least two different regular embeddings of Qn for each n > 2 has been known for some time: in [24], Nedela and Škoviera constructed a regular embedding of Qn for every solution e of the congruence e2 ≡ 1 mod n, with different solutions giving rise to non-isomorphic maps. Later, Du, Kwak and Nedela [9] J Algebr Comb (2011) 33: 215–238 217 proved that there are no other regular embeddings of Qn into orientable surfaces when n is odd. In contrast, Kwon [21] constructed new regular embeddings for all even n  6, by applying a ‘switch’ operator (as defined in [25]); he thereby also derived an exponential lower bound in terms of n for the number of non-isomorphic regular embeddings of Qn . Recently, Jing Xu [28] proved that the embeddings constructed by Kwon cover all regular embeddings of Qn into orientable surfaces, when n = 2m for odd m. In [22], Kwon and Nedela proved that there are no regular embeddings of Qn into nonorientable surfaces, for all n > 2. Also recently, the first and fifth authors of this paper gave a characterization of all orientably-regular embeddings of Qn (in terms of certain ‘quadrilateral identities’), and a construction for new regular embeddings of Qn for all n divisible by 16, not covered by the family of embeddings found by Kwon; see [3]. The aim of the present paper is to classify the regular embeddings of Qn for all n. By [22], these are orientable for n > 2, and by [9] they are known for all odd n, so we concentrate on the case where n is even, say n = 2m. In our main theorem (Theorem 5.1), we will show that when n = 2m the orientably-regular embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group Sn such that σ fixes n, preserves the set of all pairs Bi = {i, i + m} for 1 ≤ i ≤ m, and induces the same permutation on this set as the permutation Bi → Bf (i) for some additive bijection f : Zm → Zm . (Note: by additive, we mean that f (i + j ) ≡ f (i) + f (j ) mod m for all i, j ∈ Zm ; and since σ 2 is trivial, this is equivalent to f being given by f : i → ei for some square root e of 1 in Zm (namely e = f (1)).) In particular, it follows that every regular embedding of Qn belongs to one of the families constructed by Kwon [21] and Catalano and Nedela [3]. This also gives rise to formulae for the numbers of embeddings that are reflexible and chiral, respectively, which show that the ratio of reflexible to chiral embeddings tends to zero for large even n. Before proving our main theorem in Sect. 5, we give some further background in Sects. 2 and 3, and introduce a reduction process in Sect. 4. Reflexibility and the enumeration formulae are then considered in Sect. 6, and the genera and other properties of the resulting maps are dealt with in Sect. 7. 2 Further background Let M be an orientably-regular map, and let G = Auto (M) be the group of all orientation-preserving automorphisms of M. Then G acts transitively on vertices, on edges, and on faces of M; in particular, every face has the same size k, say, and every vertex has the same degree (valency) m, say. The pair {k, m} is then called the type of the map M. Moreover, for any given flag (v, e, f ) of M, there exists an automorphism R in G inducing a single-step rotation of the edges incident with f , and an automorphism S in G inducing a single-step rotation of the edges incident with v, with product RS an involutory automorphism that reverses the edge e. By connectedness, R and S generate G, which is therefore a quotient of the ordinary (k, m, 2) triangle group 218 J Algebr Comb (2011) 33: 215–238 o (k, m, 2) =  x, y | x k = y m = (xy)2 = 1 (under an epimorphism taking x to R and y to S). The map M is reflexible if and only if the group G admits an automorphism of order 2 taking R to R −1 and S to S −1 , or equivalently (by conjugation), an automorphism of order 2 taking S to S −1 and RS to S −1 R −1 = (RS)−1 = RS. Conversely, given any epimorphism θ from o (k, m, 2) to a finite group G with torsion-free kernel, a map M can be constructed using (right) cosets of the images of x, y and xy as vertices, faces and edges, with incidence given by non-empty intersection, and then G acts regularly on the arcs of M by (right) multiplication. From this point of view the study of regular maps is simply the study of smooth finite quotients of triangle groups, with ‘smooth’ here meaning that the orders of the elements x, y and xy are preserved. An isomorphism between maps is an isomorphism between their underlying graphs that preserves oriented faces. Isomorphic regular maps have the same type, and therefore come from the same triangle group; in fact, two orientably-regular maps of the same type {k, m} are isomorphic if and only if they are obtainable from the same torsion-free normal subgroup of o (k, m, 2). Rotary maps can be classified according to the genus or the Euler characteristic of the supporting surface, or by the underlying graph, or by the automorphism group of the map. Deep connections exist between maps and other branches of mathematics, which go far beyond group theory, and include hyperbolic geometry, Riemann surfaces and, rather surprisingly, number fields and Galois theory, based on observations made by Belyı̆ and Grothendieck; see [18] for example. The correspondence between rotary maps and normal subgroups of finite index in triangle groups has been exploited to develop the theory of such maps and produce or classify many families of examples. In particular, it was used by Conder and Dobcsányi in [5] to determine all rotary maps of Euler characteristic −1 to −28 inclusive, and subsequently extended by Conder in [4] for characteristic −1 to −200. Now we turn to the cube graphs Qn . For each integer n > 1, the n-dimensional cube graph Qn is the graph on vertex-set V = Z2 n , with two vertices u, v ∈ V adjacent if and only if the Hamming distance d(u, v) between them is 1 (that is, if and only if u and v differ in exactly one coordinate position). The automorphism group of Qn is well known to be the wreath product Z2 ≀ Sn , which is a semi-direct product Z2 n ⋊ Sn of V = Z2 n by the symmetric group Sn . In particular, we may view any element of Aut(Qn ) as a product of some v ∈ V with a permutation π ∈ Sn , and multiplication follows from the rule vπ = πv π where v π denotes the vector in V obtained from v by applying the permutation π to the coordinates of v. In any orientably-regular embedding of Qn , we may choose the rotation S about the vertex v = 0 to be the n-cycle ρ = (1, 2, 3, . . . , n) in Sn , and then choose the rotation R about a face f incident with v so that RS is the involution en σ , where en = (0, 0, . . . , 0, 1) is the nth standard basis vector for V , and σ is a permutation of order 1 or 2 in Sn fixing n. This is explained further in [21], where e0 is used in place of en for the purposes of consistency with taking residues modulo n. For any such permutation σ (of order 1 or 2 and fixing n) in Sn , let G(σ ) = ρ, en σ . By [21, Lemma 3.1], this subgroup of Aut(Qn ) acts transitively on the arcs of Qn . Next, if G(σ ) acts regularly on the arcs of Qn , so that |G(σ )| = n2n , J Algebr Comb (2011) 33: 215–238 219 then we call the permutation σ an admissible involution (allowing an ‘involution’ to have order 1), and we denote the corresponding regular embedding by M(σ ). In particular, the identity permutation ι is an admissible involution in Sn , giving the standard embedding M(ι). We can now state the following theorem. Theorem 2.1 (Kwon [21, Theorem 3.1]) Every regular embedding of Qn is isomorphic to M(σ ) for some admissible involution σ ∈ Sn . Moreover, for any admissible involutions σ1 , σ2 ∈ Sn , the maps M(σ1 ) and M(σ2 ) are isomorphic if and only if σ1 = σ2 . Hence the classification of regular embeddings of Qn is equivalent to the classification of admissible involutions σ in Sn . We remark that for n = 2 the standard embedding is the only regular orientable embedding of Q2 , and so from now on, we suppose n > 2. For some time it has been known (see [24], for example) that for every square root e of 1 in Zn , the mapping τe : Zn → Zn given by τe : i → ei (multiplication by e) gives rise to an admissible involution in Sn (when we think of 0 as n). The classification of regular embeddings of Qn for n odd was achieved by proving the following: Theorem 2.2 (Du, Kwak & Nedela [9]) If n is odd and σ ∈ Sn is an admissible involution, then σ = τe for some e satisfying e2 ≡ 1 mod n. In this paper we focus attention on the even-dimensional case. In this case, the following partial results are known: Theorem 2.3 (Kwon [21, Theorems 4.1 & 5.2)] For n = 2m (even), let e be a square root of 1 in Zn , and let χA be the characteristic function of a subset A ⊆ Zn \ {0} preserved by τe , ρ m , where ρ = (1, 2, . . . , n). Then the mapping σ : Zn → Zn given by σ : i → ei + mχA (i) (K) gives an admissible involution in Sn . Theorem 2.4 (Catalano & Nedela [3, Theorem 5.3]) For n = 2m where m is divisible by 8, let e be a square root of m + 1 in Zn , and let χA be the characteristic function of a subset A ⊆ Zn \ {0} such that χA (i + m) = χA (i) and χA (ei) ≡ χA (i) + i mod 2 for all i ∈ Zn . Then the mapping σ : Zn → Zn given by σ : i → ei + mχA (i) (CN) is an admissible involution. Admissible involutions defined by (K) and (CN) may be called K-involutions and CN-involutions, respectively. Jing Xu extended the classification for n odd to the case n = 2m where m is odd, by proving the following: 220 J Algebr Comb (2011) 33: 215–238 Theorem 2.5 (Xu [28]) Let n = 2m where m is odd. Then an involution σ in Sn fixing n is admissible if and only if it is a K-involution. One may observe that any K- or CN-involution commutes with ρ m , when n = 2m. In fact, this holds for any admissible involution: Proposition 2.6 Let H be a permutation group of even degree 2m containing a regular element y (acting as a 2m-cycle), such that the stabilizer of each point is a 2-group. Then y m is central in H , so the m orbits of y m  form a system of imprimitivity for H . Proof We prove this by induction on m. If m = 1 then the result is trivial. Now suppose m > 1. The lengths of orbits of a point-stabilizer Hα are powers of 2, so the fixed point set P of Hα must have even size. If |P | = 2m, then H = y and the result is immediate. If not, then P is a block of imprimitivity for H , and the action of the setwise stabilizer H{P } on P satisfies the hypotheses, with y 2m/|P | acting regularly, so that by induction, we may assume that y m is central in H{P } and that the orbits of y m  on P form a system of imprimitivity for H{P } . It then follows that the translates of those orbits form a system of imprimitivity for H . As y m induces a 2-cycle on each such block, y m is central in H .  Corollary 2.7 If σ is any admissible involution in S2m , then σ commutes with ρ m , and the orbits {i, i + m} of ρ m  form a system of imprimitivity for ρ, σ . 3 Some technical observations Let {e1 , e2 , . . . , en } be the standard orthonormal basis for V = Z2 n , and for any subset J of {1, 2, . . . , n}, let eJ be the characteristic vector of J (so that e{i} = ei for all i). Then multiplication in Z2 ≀ Sn ∼ = V ⋊ Sn is given by (eJ π)(eK μ) = eL πμ for J, K ⊆ {1, 2, . . . , n} and π, μ ∈ Sn , −1 where L is the symmetric difference of J and K π . Now suppose σ is an admissible involution in Sn , so G(σ ) = ρ, en σ  has order n2n . For 1 ≤ i ≤ n, conjugating en σ by powers of ρ gives ρ −i (en σ )ρ i = ei (ρ −i σρ i ) as an element of G(σ ), and the above multiplication then gives elements in G(σ ) of the form eL θ for every subset L of {1, 2, . . . , n}. Furthermore, post-multiplication by powers of ρ gives at least n possibilities for the element θ in Sn , for each subset L. In fact since |G(σ )| = n2n , we have the following: Lemma 3.1 If σ is an admissible involution in Sn , then for each L ⊆ {1, 2, . . . , n}, the set of all elements of G(σ ) of the form eL π for π ∈ Sn is a left coset of ρ, of size n. In particular, for each i ∈ {1, 2, . . . , n} there is a unique permutation γi ∈ Sn fixing n such that ei γi ∈ G(σ ). Clearly γn = σ , and more generally, since G(σ ) contains σ ρ −i (en σ )ρ i = ei (ρ −i σρ i ), we find that γi = ρ −i σρ −(−i) for 1 ≤ i ≤ n. J Algebr Comb (2011) 33: 215–238 221 This leads to an alternative proof of the quadrilateral identities given in [3], involving the permutation τ in Sn induced by multiplication by −1 in Zn : Proposition 3.2 If σ is an admissible involution in Sn , then στ σρ j σρ j σρ j (σ τ )2 σρ j (σ τ )3 = 1 for all j ∈ Zn . (∗) Proof First note that if i ∈ {1, 2, . . . , n} and i γn = i σ = ℓ then (ei γi )(en γn ) = ei en γi γn while (en γn )(eℓ γℓ ) = en ei γn γℓ . But ei en = en ei since V is Abelian, and γi γn and γn γℓ both fix n, so we deduce that γi γ n = γn γℓ whenever ℓ = i σ . σ σ σ Now γi γn = ρ −i σρ −(−i) σ while γn γℓ = σρ −ℓ σρ −(−ℓ) = σρ −i σρ −(−i hence σ )σ , and   (σ τ )2 στ σ σ τσ σ σ 1 = (γi γn )−1 γn γℓ = σρ (−i) σρ i σρ −i σρ −(−i ) = σρ i σρ i σρ i σρ i . Taking i = j σ τ (or, equivalently, j = i τ σ ) gives the required identity.  Corollary 3.3 If σ is an admissible involution in Sn , then (σ τ )4 = 1. στ Proof Take j = k σ τ in the above identity, to obtain σρ k σρ k and put this together with all k. (σ τ ) στ σρ k σρ k σρ k 2 (σ τ ) σρ k (σ τ )2 σρ k 3 = 1, to give (σ τ )3 σρ k (σ τ ) ρk (σ τ )4 4 = ρk = 1, for  The converse of Proposition 3.2 holds as well. This was shown in [3], but again we give an alternative proof (below). Proposition 3.4 If σ is an involution in Sn that fixes n and satisfies the quadrilateral identities (∗), then σ is admissible. Proof We prove that G(σ ) = ρ, en σ  has order n2n , by showing it contains a unique left coset of the form eL γL ρ with γL ∈ Sn fixing n, for every L ⊆ {1, 2, . . . , n}. σ Define γi = ρ −i σρ −(−i) for 1 ≤ i ≤ n, as previously. Then each γi is an element σ of Sn fixing n such that ei γi = ρ −i (en σ )ρ i ρ −(−i) −i lies in G(σ ). Moreover, since G(σ ) = ρ, en σ , every element w of G(σ ) can be expressed as a product of conjugates of en σ by powers of ρ, followed by some power of ρ, and hence has the form w = ei1 γi1 ei2 γi2 · · · eir γir ρ s for some i1 , i2 , . . . , ir and s. The multiplication rule (ea γa )(eb γb ) = (ea ec )γa γb whenever b = cγa can then be used to rewrite w in the form w = eL γi1 γi2 · · · γir ρ s for some L ⊆ {1, 2, . . . , n}. 222 J Algebr Comb (2011) 33: 215–238 The quadrilateral identities (∗) imply that for given L, the element γi1 γi2 · · · γir is uniquely determined. To see this, note that if b = cγa and d = a γc , then the above multiplication rule gives (ea γa )(eb γb ) = (ea ec )γa γb while (ec γc )(ed γd ) = (ec ea )γc γd . Since ea ec = ec ea , all we have to do is to prove that γa γb = γc γd whenever b = cγa = −c −(−c)σ −a −(−a)σ = (a − c)σ − (−c)σ . = (c − a)σ − (−a)σ and d = a γc = a ρ σρ cρ σρ σ The quadrilateral identity for j = (a − c) is στ σ 1 = σρ (a−c) σρ c−a σρ (c−a) σρ (c−a) σ which can be rewritten as 1 = σρ d+(−c) σρ c−a σρ −(−a) jugation this becomes 1 = ρ −a σρ −(−a) σ −b σρ (c−a) (σ τ )2 (σ τ )2 σ −b , σρ (c−a) (σ τ )2 . Upon con- σ σρ d+(−c) σρ c , 2 which can be rewritten as 1 = γa γb ρ u γd−1 γc−1 where u = (−b)σ + (c − a)(σ τ ) − (−d)σ . Thus (γa γb )−1 γc γd = ρ u , and as the left-hand side of this identity fixes n, we find ρ u = 1, so (γa γb )−1 γc γd = 1 and therefore γa γb = γc γd , as required.  Corollary 3.5 Let σ be any involution in S2m such that σ fixes n = 2m, preserves the set of all pairs Bi = {i, i + m} for 1 ≤ i ≤ m, and induces the same permutation on this set as the permutation Bi → Bf (i) for some additive bijection f : Zm → Zm . Then σ is admissible. Proof It is an easy exercise to verify that σ satisfies the quadrilateral identities (∗).  Note that the condition that σ preserves the set {Bi : 1 ≤ i ≤ m} is equivalent to σ commuting with ρ m = (1, m + 1)(2, m + 2) · · · (m, 2m). 4 Reduction In this section we describe a reduction from the case of Qn to the case of Qm when n = 2m (even). This can be used to provide an alternative proof of Theorem 2.5, as well as assist with the proof of our main theorem in the next section. To do this, we consider the natural action of the wreath product Z2 ≀ Sn on the set {1, 2, . . . , 2n}, with block-set {{i, i + n} : 1 ≤ i ≤ n} preserved by V = Z2 n and permuted by Sn . Indeed let ei induce the transposition (i, i + n) for 1 ≤ i ≤ n, and let ρ induce the permutation (1, 2, . . . , n)(n + 1, n + 2, . . . , 2n). Lemma 4.1 Suppose n = 2m, and σ is an admissible involution in Sn . Let K be the subgroup of Z2 ≀ Sn generated by ρ m and ei ei+m for 1 ≤ i ≤ m. Then K is an Abelian subgroup of G(σ ), of order 2m+1 . Moreover, K is a normal subgroup of G(σ ), and consists of all elements that preserve each of the sets Pi = {{i, i + m}, {i + 2m, i + 3m}} (and each of the sets Qi = {{i, i + 3m}, {i + 2m, i + m}}), for 1 ≤ i ≤ m. J Algebr Comb (2011) 33: 215–238 223 σ Proof First, let G = G(σ ), and let γj = ρ −j σρ −(−j ) be the elements defined in Sect. 3. By Corollary 2.7, we know that σ permutes the sets Bi = {i, i + m} among themselves, and hence that (i + m)σ = i σ + m (mod n) for all i. Then since ρ m commutes with σ , we find that σ γi+m = ρ −(i+m) σρ −(−(i+m)) = ρ −i ρ −m σρ m ρ −(−i) σ = ρ −i σρ −(−i) = γi σ for 1 ≤ i ≤ m. In particular, as G contains ei γi and ei+m γi+m = ei+m γi , it follows that G contains (ei γi )(ei+m γi )−1 = ei ei+m for all i, so K is a subgroup of G. Also the generators of K are commuting involutions, so K is Abelian, of order 2m+1 . Observe that both ρ and σ centralize ρ m and conjugate the ei ei+m among themselves, while en centralizes all the ei ei+m and conjugates ρ m to em e2m ρ m . It follows that K is normalized by each of ρ, σ and en , and in particular, K is normal in ρ, en σ  = G. Next, let H be the stabilizer in G of the two points m and 2m (or equivalently, of the four points m, 2m, 3m and 4m). Since the stabilizer in G of m fixes 3m and has {2m, 4m} as one of its orbits, and has index 2n = 4m in G, this subgroup H has index 4n in G, so has order 2n−2 . Now consider the subgroup H K. The intersection H ∩ K contains all the ei ei+m for i = m, 2m, but does not contain em e2m , ρ m or em e2m ρ m (which take m to 3m, 2m and 4m respectively), so H ∩ K has index 4 in K and therefore has order 2m−1 . Thus |H K| = |H ||K|/|H ∩ K| = 2n−2+m+1 /2m−1 = 2n , so the index of H K in G is n = 2m. It follows that H K is the stabilizer in G of the set Pm = {{m, 2m}, {3m, 4m}}, the images of which under other elements of G are the sets Pi and Qi given in the statement of this Lemma. Moreover, the core of H in G is trivial (being the stabilizer of all points), so the core of H K in G is K. This completes the proof.  The above lemma gives a quotient G(σ )/K that acts transitively on a set of size 2m, namely the set of all Pi and Qi . The permutation induced by ρ is a pair of m-cycles, namely (P1 , P2 , . . . , Pm ) and (Q1 , Q2 , . . . , Qm ). But also the generators ei of V = Z2 n and the involution σ induce permutations of this set, with each ei interchanging the points Pi and Qi while fixing all others, and σ inducing effectively the same permutation on the Qi as it does on the Pi . In particular, since σ commutes with ρ m , the orbits {Pi , Pi+m } and {Qi , Qi+m } of ρ m  form a system of imprimitivity for G(σ )/K, which accordingly can be viewed as a subgroup of the wreath product Z2 ≀ Sm . Furthermore, we may note that σ fixes Qm (and Pm ), and hence that en σ interchanges Pm and Qm while otherwise acting to preserve the sets {P1 , P2 , . . . , Pm−1 } and {Q1 , Q2 , . . . , Qm−1 }. In other words, K is the kernel of a reduction, from G(σ ) as a subgroup of Z2 ≀ Sn , to G(σ )/K which is a subgroup of Z2 ≀ Sm in its natural action on the Pi and Qi (with {Pi , Qi } as the ‘base pairs’). In particular, G/K has order 2m m, and is the group of orientation-preserving automorphisms of a regular embedding of Qm . In fact this permutation induced by σ is effectively the same as the one induced by σ on the blocks Bi = {i, i + m} of the natural action of ρ, σ  on {1, 2, . . . , n}. This gives another way of defining the reduction. As explained in [3], we may directly define the projections ρ and σ of ρ and σ in Sm by letting i ρ and i σ be the 224 J Algebr Comb (2011) 33: 215–238 residues mod m of i ρ and i σ respectively, for 1 ≤ i ≤ m. Then σ obviously satisfies the quadrilateral identities, and is therefore an admissible involution in Sm . Reciprocally, we may call σ an admissible lift of σ . By the above remarks, we now have the following: Proposition 4.2 Every admissible involution σ ∈ S2m is an admissible lift of some admissible involution in Sm . Note that every K-involution and every CN-involution in S2m is an admissible lift of the involution τe : Zm → Zm given by multiplication by some square root e of 1 in Zm . This was observed in [3], where it was also proved that every admissible lift of such an involution τe in Sm is a K-involution or CN-involution in S2m ; see [3, Theorem 5.3]. We also now have the following: Alternative proof of Theorem 2.5 For n = 2m where m is odd, let σ be an admissible involution in Sn . By the above reduction, σ is an admissible involution in Sm , so by Theorem 2.2, we know that σ = τe for some square root e of 1 in Zm . Now replace e by e + m if e is even. Then e2 ≡ 1 mod 2 and mod m, so e2 ≡ 1 mod n. Taking A = {i ∈ Zn : i σ = ei (mod n)} = {i ∈ Zn : i σ = ei + m (mod n)}, we see that 0 ∈ A and that A is preserved by both ρ m and multiplication by e mod n, so σ is a K-involution.  5 Classification theorem In this section, we give a characterization of all admissible involutions in S2m , for every positive integer m. When taken together with Theorem 2.2, this gives a complete classification of all regular embeddings of hypercubes Qn . Theorem 5.1 Let n = 2m be an even positive integer, and let ρ = (1, 2, 3, . . . , n) in Sn . Then every regular embedding of Qn is isomorphic to the embedding M(σ ) for some permutation σ of order 1 or 2 in Sn and fixing n, such that: (1) σ commutes with ρ m , so that the sets Bi = {i, i + m} (for 1 ≤ i ≤ m) form a system of imprimitivity for ρ, σ  on {1, 2, . . . , n}, and (2) σ permutes the blocks Bi in the same way as the permutation Bi → Bf (i) for some additive bijection f : Zm → Zm . Moreover, every such σ gives a regular embedding of Qn , and distinct σ give nonisomorphic embeddings. Part (1) follows from Corollary 2.7, and we will prove part (2) by induction on m. We have already seen that when m is odd, this follows from Theorem 2.5, so we suppose that m is even, say m = 2k, and let σ be any admissible involution in S2m . By the reduction described in Sect. 4, we know that the action of σ on the blocks Bi is the same as that of an admissible involution σ in Sm , and now by induction, we may assume that the projection of σ in Sk is an additive bijection from Zk to Zk . J Algebr Comb (2011) 33: 215–238 225 Let e = 1σ if this is odd, or otherwise let e = 1σ + k (which will be odd, since k is odd when 1σ is even). Then by additivity of the projection of σ in Sk , we can prove by induction that i σ ≡ ei mod k for all i ∈ Zn . Hence we may define a function ψ : Zn → Z4 satisfying i σ = ei + kψ(i) for all i ∈ Zn . The remainder of our proof will depend heavily on properties of this function ψ and related objects. Lemma 5.2 If e ∈ Zn and ψ : Zn → Z4 are defined as above, then: (a) (b) (c) (d) (e) ψ(0) = ψ(m) = 0, and ψ(k) and ψ(3k) are both even; e2 ≡ δk + 1 mod n for some δ ∈ Z4 ; δi + eψ(i) + ψ(i σ ) ≡ 0 mod 4, for all i ∈ Zn ; ψ(i + m) = ψ(i) for all i ∈ Zn ; ψ(i + k) ≡ ψ(i) mod 2 for all i ∈ Zn . Proof Parts (a) and (b) are obvious from the definitions. For part (c), observe that  σ          i = i σ = ei σ + kψ i σ = e ei + kψ(i) + kψ i σ = i + k δi + eψ(i) + ψ i σ in Zn , since e2 = 1 + kδ by part (b). Part (d) is a consequence of the fact that σ commutes with ρ m :     0 = (i + m)σ − i σ + m = em + k ψ(i + m) − ψ(i) − m   = k ψ(i + m) − ψ(i) . Similarly, (i + k)σ − i σ = ek + k(ψ(i + k) − ψ(i)) = k(e + ψ(i + k) − ψ(i)), and since the left-hand side is either k or 3k(mod n), and e is odd, we obtain part (e).  We wish to prove that σ is additive when reduced modulo m. Now since (i + j )σ − i σ − j σ = e(i + j ) + kψ(i + j ) − ei − kψ(i) − ej − kψ(j )   = k ψ(i + j ) − ψ(i) − ψ(j ) we can define ψ(i, j ) = ψ(i + j ) − ψ(i) − ψ(j ) in Z4 , and then it suffices to prove that ψ(i, j ) is even for all i, j ∈ Zn . We will call a pair (i, j ) good if ψ(i, j ) is even, and bad otherwise. In a sequence of further observations (Lemma 5.3 to Proposition 5.18) we will prove that there are no bad pairs, and hence σ is an admissible lift of its additive projection σ . Note here that −i σ stands for −(i σ ), rather than (−i)σ (which can differ from −(i σ )). Lemma 5.3 ψ(i σ , −i σ ) ≡ ψ(ei, −ei) ≡ ψ(i, −i) mod 2 for all i ∈ Zn . 226 J Algebr Comb (2011) 33: 215–238 Proof Since i σ = ei + kψ(i), we have ψ(i σ ) ≡ ψ(ei) mod 2 by Lemma 5.2(e), and similarly, ψ(−i σ ) ≡ ψ(−ei) mod 2. Then by Lemma 5.2(c) and since e is odd we find that ψ(ei) ≡ ψ(i σ ) ≡ −δi − eψ(i) ≡ −δi − ψ(i) mod 2, and replacing i by −i gives also ψ(−ei) ≡ δi − ψ(−i). Adding these last two congruences gives     ψ i σ + ψ −i σ ≡ ψ(ei) + ψ(−ei) ≡ −ψ(i) − ψ(−i) ≡ ψ(i) + ψ(−i) mod 2, and the rest follows since ψ(t, −t) = ψ(0) − ψ(t) − ψ(−t) = −(ψ(t) + ψ(−t)) for all t.  2 Corollary 5.4 i (σ τ ) ≡ i + kψ(i, −i) mod m for all i ∈ Zn . Proof   σ      2 i (σ τ ) = − −i σ = ei σ − kψ −i σ = e ei + kψ(i) − kψ −i σ    by Lemma 5.2(b) = i + k δi + eψ(i) − ψ −i σ  σ    σ by Lemma 5.2(c) = i + k −ψ i − ψ −i  σ σ = i + kψ i , −i , 2 and thus i (σ τ ) ≡ i + kψ(i, −i) mod m, by Lemma 5.3. Lemma 5.5  ψ(i, j ) + ψ(i + kψ(i, −i), j + kψ(i, j )) ≡ 0 mod 4 for all i, j ∈ Zn . Proof First we observe that for every t, i ∈ Zn , Lemma 5.2 gives t σρ i σρ i σ τ  σ = i + t σ − iσ     = et σ + k ψ i + t σ − ψ(i)       = e et + kψ(t) + k ψ i + t σ − ψ(i)     = t + k δt + eψ(t) + ψ i + t σ − ψ(i)       = t + k −ψ t σ + ψ i + t σ − ψ(i)   = t + kψ i, t σ . i Replacing t by t σρ σρ (∗) for t then gives iσ τ by Lemma 5.2(c) 2 and i by i (σ τ ) here, and applying the quadrilateral identity  2 i iσ τ  + kψ i (σ τ ) , t σρ σρ σ     2 i i σ τ  = t + k ψ i, t σ + ψ i (σ τ ) , t σρ σρ σ      σ  2  = t + k ψ i, t σ + ψ i (σ τ ) , t + kψ i, t σ t = t σρ by Lemma 5.2(b) i σρ i σ τ by the above. Letting t = j σ (so that also j = t σ ), we find that  σ  2  ψ(i, j ) + ψ i (σ τ ) , j σ + kψ(i, j ) ≡ 0 mod 4. J Algebr Comb (2011) 33: 215–238 227 Further application of Lemma 5.2 gives  σ σ j + kψ(i, j )     = e j σ + kψ(i, j ) + kψ j σ + kψ(i, j )     = e ej + kψ(j ) + kψ(i, j ) + kψ j σ + kψ(i, j )    = j + k δj + eψ(j ) + eψ(i, j ) + ψ j σ + kψ(i, j )       = j + k −ψ j σ + ψ j σ + kψ(i, j ) + eψ(i, j ) by Lemma 5.2(b) by Lemma 5.2(c) ≡ j + keψ(i, j ) mod m by Lemma 5.2(e) ≡ j + kψ(i, j ) mod m since e is odd, and inserting this into the previous equation (and using Lemma 5.2(d)) we obtain   2 ψ(i, j ) + ψ i (σ τ ) , j + kψ(i, j ) ≡ 0 mod 4. 2 On the other hand, by Lemma 5.4 we have i (σ τ ) ≡ i + kψ(i, −i) mod m, and so the required congruence follows from Lemma 5.2(d).  Next, by Lemma 5.2(e), we may define another function c : Zn → Z2 ψ(t + k) = ψ(t) + 2c(t) for all t ∈ Zn . It is easy to see that c(0) = c(m) = 0, and that ψ(t + kd) = ψ(t) + 2dc(t) for all d, again by parts (d) and (e) of Lemma 5.2. We use this function and the previous lemma to prove the following: Lemma 5.6 ψ(i, −i) is even, for all i ∈ Zn . Proof Assume the contrary, so that ψ(i, −i) is odd for some i. By Lemma 5.5 and the definition of c, for all i, j ∈ Zn we have   0 ≡ ψ(i, j ) + ψ i + kψ(i, −i), j + kψ(i, j )    ≡ ψ(i, j ) + ψ i + j + k ψ(i, −i) + ψ(i, j )     − ψ i + kψ(i, −i) − ψ j + kψ(i, j )   ≡ 2ψ(i, j ) + 2 ψ(i, −i) + ψ(i, j ) c(i + j ) − 2ψ(i, −i)c(i) − 2ψ(i, j )c(j ) mod 4, and hence (from the assumption that ψ(i, −i) is odd), we have   ψ(i, j ) + 1 + ψ(i, j ) c(i + j ) − c(i) − ψ(i, j )c(j ) ≡ 0 mod 2, or equivalently,     1 + ψ(i, j ) c(i + j ) ≡ c(i) + ψ(i, j ) c(j ) − 1 mod 2. 228 J Algebr Comb (2011) 33: 215–238 This can be used to prove by induction that c(t) = c(i) whenever t is a multiple of i in Zn . For if that is true for t = j , then ψ(i, j ) must be even (or otherwise (1 + ψ(i, j ))c(i + j ) would be even while c(i) + ψ(i, j )(c(j ) − 1) ≡ 2c(i) − 1 ≡ 1 mod 2); and it then follows easily that c(i +j ) ≡ (1+ψ(i, j ))c(i +j ) ≡ c(i) mod 2, so it is true also for t = i + j . But on the other hand, since c(0) = 0 and ψ(i, −i) is odd, taking j = −i in the last displayed congruence above gives 0 ≡ c(i) + c(−i) − 1 mod 2, so c(−i) = c(i), a contradiction. This completes the proof.  2 Corollary 5.7 (σ τ )2 acts trivially modulo m; that is, i (σ τ ) ≡ i mod m for all i ∈ Zn . 2 Proof We know i (σ τ ) ≡ i + kψ(i, −i) mod m, by Corollary 5.4, and the rest follows from Lemma 5.6.  Also Lemma 5.6 can be used to provide a simpler version of Lemma 5.5: Corollary 5.8 ψ(i, j )((1 + c(i + j ) − c(j )) ≡ 0 mod 2 for all i, j ∈ Zn . Proof First Lemma 5.5 gives this congruence mod 4:   0 ≡ ψ(i, j ) + ψ i + kψ(i, −i), j + kψ(i, j )    ≡ ψ(i, j ) + ψ i + j + k ψ(i, −i) + ψ(i, j )     − ψ i + kψ(i, −i) − ψ j + kψ(i, j ) . Since ψ(i, −i) is even, it follows that   0 ≡ ψ(i, j ) + ψ i + j + kψ(i, j )   − ψ(i) − ψ j + kψ(i, j ) ≡ 2ψ(i, j ) + 2ψ(i, j )c(i + j ) − 2ψ(i, j )c(j )   ≡ 2ψ(i, j ) 1 + c(i + j ) − c(j ) mod 4, by Lemma 5.2(d) by the definition of c and the result follows. Lemma 5.9 c(i + k) = c(i) and c(ei) = c(i) for all i ∈ Zn . Proof The first of these is an easy consequence of the following: ψ(i) = ψ(i + m) = ψ(i + k + k) = ψ(i + k) + 2c(i + k) = ψ(i) + 2c(i) + 2c(i + k). For the second, note that by Lemma 5.2(c) and the definitions of ψ and c, we have 0 ≡ iδ + eψ(i) + ψ(ei) + 2c(ei)ψ(i) mod 4.  J Algebr Comb (2011) 33: 215–238 229 Replacing i by i + k, we have also 0 ≡ (i + k)δ + eψ(i + k) + ψ(ei + ek) + 2c(ei + ek)ψ(i + k) mod 4. But now ψ(ei + ek) = ψ(ei) + 2ec(ei), and by what we proved above, c(ei + ek) = c(ei), so the latter congruence can be rewritten as 0 ≡ (i + k)δ + eψ(i + k) + ψ(ei) + 2ec(ei) + 2c(ei)ψ(i + k) mod 4. Subtracting the earlier congruence (namely the one for i) from this one (for i + k), and again using ψ(i + k) = ψ(i) + 2c(i), we find that   0 ≡ kδ + 2ec(i) + 2ec(ei) + 4c(ei)c(i) ≡ kδ + 2e c(i) + c(ei) mod 4. Finally, since e is odd and n is divisible by 4, we know that kδ + 1 ≡ e2 ≡ 1 mod 4, and so c(i) + c(ei) must be even.  Corollary 5.10 iδ + (e + 2c(i))ψ(i) + ψ(ei) ≡ 0 mod 4 for all i ∈ Zn . Proof We observed that 0 ≡ iδ + eψ(i) + ψ(ei) + 2c(ei)ψ(i) mod 4 in the proof of Lemma 5.9. Since c(ei) = c(i), the result follows.  Next, recall that a pair (i, j ) is good if ψ(i, j ) = ψ(i + j ) − ψ(i) − ψ(j ) is even, or equivalently, if (i + j )σ ≡ i σ + j σ mod m, and bad otherwise. Clearly (i, 0) and (0, j ) are good for all i and j . Moreover, since (i + m)σ = i σ + m = i σ + mσ , we know that (i, m) is good, for all i ∈ Zn , and it follows that (i, mj ) is good for all j . We also have the following: Lemma 5.11 If (i, j ) is a bad pair, then the pairs (j, i), (i σ , j σ ), (j σ , i σ ), (−i, −j ), (−j, −i) and (−i, i + j ) are all bad. Proof The first of these six follows from the definition, the second one from σ 2 = 1, and the third is a combination of the first two. The fourth and fifth follow from Corollary 5.7. For the last one, note that j σ ≡ (−i)σ + (i + j )σ ≡ −(i σ ) + (i + j )σ mod m.  Lemma 5.12 If (i, j ) is a bad pair, then   c(i) = c(j ) = c −(i + j ) = c(−i) = c(−j ) = c(i + j ). Equivalently, if c(u) and c(v) have opposite parities, then the pair (u, v) must be good. Proof By Lemma 5.8, we know that ψ(i, j )((1 + c(i + j ) − c(j )) ≡ 0 mod 2 for every pair (i, j ), whether good or bad. Now if (i, j ) is bad, then ψ(i, j ) is odd, and therefore 1 + c(i + j ) − c(j ) is even, so c(j ) and c(i + j ) have opposite parities. The  rest follows from Lemma 5.11. 230 J Algebr Comb (2011) 33: 215–238 Lemma 5.13 If (i, j ) is a bad pair and (u, v) a good pair, with i + j ≡ u + v mod m, then exactly one of (i, −u) and (v, −j ) is bad, and exactly one of (i, −v) and (u, −j ) is bad. Proof First i σ + j σ ≡ (i + j )σ ≡ (u + v)σ ≡ uσ + v σ mod m, so by Corollary 5.7, we find i σ + (−u)σ ≡ i σ − uσ ≡ v σ − j σ ≡ v σ + (−j )σ mod m. On the other hand, i + (−u) = v + (−j ) mod m (since i + j = u + v mod m), so (i + (−u))σ ≡ (v + (−j ))σ mod m. It follows that (i + (−u))σ ≡ i σ + (−u)σ or (v + (−j ))σ ≡ v σ + (−j )σ mod m. Just one of these holds, since i σ + (−u)σ ≡ (i − u)σ ≡ (v − j )σ ≡ v σ + (−j )σ mod k, by our assumption that σ is additive modulo k. Hence exactly one of (i, −u) and (v, −j ) is bad. Similarly, exactly one of (i, −v) and (u, −j ) is bad.  Lemma 5.14 If (i, j ) is bad, then for every positive integer a, there exists some v ∈ Zn such that the pair (2a i, v) is bad. Proof By Lemma 5.12, we know c(i) = c(i + j ) and hence the pair (i, i + j ) is good. Also (−i, 2i + j ) must be good, for otherwise (i, i + j ) would be bad, by Lemma 5.11. Now since (i, j ) is bad and (−i, 2i + j ) is good, Lemma 5.13 shows that exactly one of (i, i) and (2i + j, −j ) is bad. In the former case, both (−i, 2i) and (2i, −i) are bad (by Lemma 5.11), while in the latter case, both (−(2i + j ), 2i) and (2i, −(2i + j )) are bad. In each case, we find that (2i, t) is bad for some t (namely −i or −(2i + j ), respectively). But now the same argument applied to (2i, t) shows that (4i, u) is bad for some u, and so on, and hence by induction, we get the result claimed.  Now let q be the largest odd divisor of k, so that n = 2m = 4k = 2s q for some s ≥ 2, and let d = gcd(e − 1, q). Lemma 5.15 With d = gcd(e − 1, q) defined as above: (a) (b) (c) (d) (e) (f) the pair (aq, b) is good for all a, b ∈ Zn ; if (i, j ) is a bad pair, then so is (i + aq, j + bq) for all a, b ∈ Zn ; the pair (a(e − 1), b) is good for all a, b ∈ Z2m ; if (i, j ) is a bad pair, then so is (i + a(e − 1), j + b(e − 1)) for all a, b ∈ Zn ; if (i, j ) is a bad pair, then so is (i + ad, j + bd) for all a, b ∈ Zn ; for all a, b ∈ Z2m , the pair (ad, b) is good. Proof For part (a), note that if (aq, b) is bad, then by Lemma 5.14, the pair (2s aq, v) is bad for some v. But 2s aq = an = 0 in Zn , so this says (0, v) is bad, contradiction. Next, if (i, j ) is bad, then by Lemma 5.11 the pair (i + j, −j ) is bad, and by (a), we know that (−aq, i + aq) is good. Hence by Lemma 5.13, we find that exactly one of (i + j, aq) and (i + aq, j ) is bad. But (i + j, aq) is good, so (i + aq, j ) must be bad. A similar argument then shows that (i + aq, j + bq) is bad, which proves (b). For part (c), let v = a(e − 1). Then   ve = a(e − 1)e = a e2 − e ≡ a(1 + kδ − e) ≡ a(1 − e) = −v mod k, J Algebr Comb (2011) 33: 215–238 231 so c(v) = c(ev) = c(−v) by Lemma 5.9, and it follows from Lemma 5.12 that no pair (v, b) is bad. Hence (a(e − 1), b) is good for all b, proving part (c). By Lemma 5.11, also (−a(e − 1), u) is good for all u. In particular, if (i, j ) is bad, then we can take u = i + j + a(e − 1), which gives a pair with the same sum as (i, j ), and by Lemma 5.13, we find that exactly one of (i, a(e − 1)) and (u, −j ) is bad. But (i, a(e − 1)) is good by part (c), so (u, −j ) must be bad, and therefore (i + a(e − 1), j ) = (u − j, j ) is bad. A similar argument then shows that (i + a(e − 1), j + b(e − 1)) is bad, which proves (d). By Bézout’s identity, d = gcd(e − 1, q) = u(e − 1) + vq for some integers u and v, and thus ad = au(e − 1) + avq ≡ au(e − 1) mod q and similarly bd ≡ bu(e − 1) mod q, for given a, b ∈ Zn . Hence if (i, j ) is bad, then (i + ad, j + bd) ≡ (i + au(e − 1), j + bu(e − 1)) mod q, so (i + ad, j + bd) is bad by parts (b) and (d). This proves (e). Finally, for part (f), if (ad, b) is bad, then by part (e), so is (ad − ad, b) = (0, b), a contradiction.  Lemma 5.16 For every integer i, there exists an integer a such that (e − 1)(i + ad) is divisible by m. Proof First, write q = du and e − 1 = d2r v, where u and v are odd integers (and r is a non-negative integer), and gcd(u, 2r v) = gcd(q/d, (e − 1)/d) = 1. Then since (e − 1)(e + 1) = e2 − 1 ≡ 0 mod k(= 2s−2 q), we know that du = q divides (e − 1)(e + 1), and as d divides e − 1 but u is coprime to 2r v = (e − 1)/d, we deduce that u divides e + 1. It follows that gcd(d, u) divides gcd(e − 1, e + 1) = 2, and since both d and u are odd, we must have gcd(d, u) = 1. But also u and v are coprime (since gcd(u, 2r v) = 1), therefore also gcd(dv, u) = 1. Thus gcd(e − 1, m/d) = gcd(d2r v, 2s−1 q/d) = gcd(d2r v, 2s−1 u) is a power of 2, say gcd(e − 1, m/d) = 2w . Now e − 1 is divisible by both 2w (= gcd(e − 1, m/d)) and d(= gcd(q, e − 1)), which is odd, so e − 1 = 2w dt for some t. Also, by Bézout’s identity, there exist integers A and B such that 2w = (e − 1)A + (m/d)B, and therefore e − 1 = 2w dt = (e − 1)Adt + mBt. For any given i, it follows that (e − 1)i = (e − 1)Adti + mBti, and hence that (e − 1)(i − (Ati)d) = mBti. Taking a = −(Ati), we have (e − 1)(i + ad) ≡ 0 mod m.  Corollary 5.17 For every integer i, there exists an integer t such that t ≡ i mod d and et ≡ t mod m. Proof By Lemma 5.16, there exists some a for which (e − 1)(i + ad) is divisible by m. Let t = i + ad. Then t ≡ i mod d, and et − t = (e − 1)t = 0 mod m.  Proposition 5.18 There are no bad pairs. Proof Suppose there exists a bad pair (i, j ). By Lemma 5.15(e), we can replace i by any integer t congruent to i mod d, and so by Corollary 5.17, we may assume that ei ≡ i mod m. Similarly, we may assume that ej ≡ j mod m. Then by Lemma 5.2(d), we find that ψ(ei) = ψ(i) and ψ(ej ) = ψ(j ). 232 J Algebr Comb (2011) 33: 215–238 Next, since ψ(i, j ) = ψ(i + j ) − ψ(i) − ψ(j ) is odd, we know that at least one of ψ(i), ψ(j ) and ψ(i + j ) is odd, and without loss of generality (replacing (i, j ) by (j, i) or (i + j, −i) if necessary), we may assume that ψ(i) is odd. We also know that c(i) = c(−i), by Lemma 5.12, so that c(−i) ≡ c(i) + 1 mod 2. Now by Corollary 5.10 and the fact that ψ(ei) = ψ(i), we find that     0 ≡ iδ + e + 2c(i) ψ(i) + ψ(ei) ≡ iδ + e + 2c(i) + 1 ψ(i) mod 4. Similarly, replacing i by −i (and using c(−i) ≡ c(i) + 1 mod 2), we have     0 ≡ −iδ + e + 2c(−i) ψ(−i) + ψ(−ei) ≡ −iδ + e + 2c(i) + 3 ψ(−i) mod 4. Adding these two congruences gives    0 = e + 2c(i) + 3 ψ(i) + ψ(−i) − 2ψ(i) mod 4. Since both e + 3 + 2c(i) and ψ(i) + ψ(−i) = ψ(i, −i) are even, their product is divisible by 4. Thus 2ψ(i) is divisible by 4, which is a contradiction.  This completes the proof of Theorem 5.1. 6 Reflexibility and enumeration In this section, we consider reflexibility of the orientably-regular embeddings M(σ ) of Qn , and then derive formulae for the total number of non-isomorphic embeddings as well as for those that are reflexible and chiral, respectively. We begin with the following: Proposition 6.1 Let σ be an admissible involution in Sn . Then the embedding M(σ ) is reflexible if and only if (σ τ )2 = 1, where τ is the permutation Sn induced by multiplication by −1 in Zn . Proof By the background theory of regular maps given in Sect. 2, we know M(σ ) is reflexible if and only if there exists an involutory automorphism θ of G(σ ) = ρ, en σ  that inverts ρ and fixes en σ . This reflecting automorphism must induce an automorphism of the underlying graph Qn , and hence can be assumed to be an element of Z2 ≀ Sn , say θ = vπ for some v ∈ V = Z2 n and π ∈ Sn . Now   π ρ −1 = ρ θ = ρ vπ = (vρv)π = v ρvρ −1 ρ π , which implies that ρvρ −1 = v and ρ π = ρ −1 . The latter implies π = τρ i for some i, and the former implies that v is either trivial (zero) or the product e1 e2 · · · en . Since e1 e2 · · · en is central in Z2 ≀Sn , we may assume without loss of generality that v is trivi i ial, so θ = τρ i for some i. If i = 0 (mod n), however, then en θ = en τρ = en ρ = ei , so (en σ )θ = ei σ θ , so θ does not centralize en σ . Thus θ = τ , which centralizes en , and then since en σ = (en σ )τ = en σ τ , the reflexibility condition reduces to requiring  σ τ = σ , or equivalently, (σ τ )2 = 1. J Algebr Comb (2011) 33: 215–238 233 Next, we consider the total number of non-isomorphic regular embeddings of Qn , or equivalently, the number of admissible involutions in Sn . We also determine how many of these embeddings are reflexible. To do this, it helps to define Inv(n) = {e ∈ Zn | e2 = 1 mod n} (the set of all square roots of 1 in Zn ) for each positive integer n. Note also that when n is an odd primepower, there are just two such roots, viz. 1 and n − 1, since the group of units in Zn is cyclic in that case. Now for odd n, by Theorem 2.2 (taken from [9]) the total number of embeddings is simply the number of square roots of 1 in Zn . If n = p1a1 p2a2 · · · ptat is the primepower decomposition of n (with p1 , p2 , . . . , pt distinct odd primes and a1 , a2 , . . . , at positive integers), this number is 2t (by the Chinese Remainder Theorem). Moreover, every admissible involution τe commutes with τ , so every regular embedding is reflexible in this case. In other words, none of the regular embeddings of Qn is chiral when n is odd. For even n, by Theorem 5.1 the total number of regular embeddings is equal to the number of permutations σ of order 1 or 2 in Sn that fix n and reduce modulo m to multiplication by some square root e of 1 in Zm . In this case, we have the following counting theorem. Theorem 6.2 For n = 2m (even), the total number of regular embeddings of Qn is  1 2 2 (m+gcd(e−1,m)−2) . e∈Inv(m) Proof Let σ ∈ Sn be an admissible involution that reduces to τe modulo m. If τe fixes i ∈ Zm \ {0}, then σ either fixes or interchanges the two points i and i + m. Similarly, if τe moves i ∈ Zm , then σ induces either (i, ei)(i + m, ei + m) or (i, ei + m)(i + m, ei) on the 4-point set {i, ei, i + m, ei + m} (considered mod n). Hence the number of admissible σ ∈ Sn that reduce to τe modulo m is 2d , where d is the number of cycles of the permutation τe on Zm \ {0}. Now i ∈ Zm is fixed by τe if and only if (e − 1)i = ei − i ≡ 0 mod m, which occurs if and only if i is divisible by m/ gcd(e − 1, m), so the number of i ∈ Zm fixed by τe is exactly gcd(e − 1, m). Hence the number of cycles of τe on Zm is d + 1 = gcd(e − 1, m) + and the result follows.  1  1 m − gcd(e − 1, m) = m + gcd(e − 1, m) , 2 2  By Lemma 6.1, the reflexible embeddings come from the admissible involutions that commute with τ , and for these we have the following: Theorem 6.3 For n = 2m (even), the number of reflexible regular embeddings of Qn is  1 4 (m+gcd(e−1,m)+gcd(e+1,m)−3) if m is odd e∈Inv(m) 2  1 4 (m+gcd(e−1,m)+gcd(e+1,m)−2) if m is even. e∈Inv(m) 2 234 J Algebr Comb (2011) 33: 215–238 Proof Let σ ∈ Sn be an admissible involution that reduces to τe modulo m, and commutes with τ (so that (−i)σ = −(i σ ) for every i ∈ Zn ). Note that σ fixes m and n. If m is even, then τe fixes m2 , and σ either fixes or interchanges the points m2 and 3m 2 . If τe fixes i ∈ Zm \ {0, m2 , m, 3m 2 }, then σ induces either the identity permutation or (i, i + m)(m − i, n − i) on the 4-point set {i, i + m, m − i, n − i} (considered mod n). Similarly, if τe takes i ∈ Zm \ {0, m2 , m, 3m 2 } to m − i (mod m), then σ induces either (i, m − i)(i + m, n − i) or (i, n − i)(i + m, m − i) on the 4-point set {i, m − i, i + m, n − i} (considered mod n). For any other i ∈ Zn (neither fixed by τe nor taken to m − i by τe ), it is easy to see that σ induces either (i, ei)(i + m, ei + m)(m − i, m − ei)(n − i, n − ei) or (i, ei + m)(i + m, ei)(m − i, n − ei)(n − i, m − ei) on the set {i, ei, i + m, ei + m, m − i, m − ei, n − i, n − ei} (considered mod n). Since the number of fixed points of τe on Zm is gcd(e − 1, m) while (similarly) the number of i ∈ Zm satisfying ei = m − i mod gcd(e + 1, m), we find the total number of possibilities for σ is 2d , where  1 gcd(e − 1, m) + gcd(e + 1, m) − 2 2  1 + m − gcd(e − 1, m) − gcd(e + 1, m) + 1 4  1 = m + gcd(e − 1, m) + gcd(e + 1, m) − 3 if m is odd, while 4  1 d = gcd(e − 1, m) + gcd(e + 1, m) − 4 2  1 + m − gcd(e − 1, m) − gcd(e + 1, m) + 2 + 1 4   1 = m + gcd(e − 1, m) + gcd(e + 1, m) − 2 if m is even, 4 d= and the result follows.  Corollary 6.4 For n = 2m (even), the number of (orientably) regular embeddings of Qn that are chiral is  1 1 2 (m+gcd(e−1,m)−2) − 2 4 (m+gcd(e−1,m)+gcd(e+1,m)−3) ) if m is odd e∈Inv(m) (2  1 1 2 (m+gcd(e−1,m)−2) − 2 4 (m+gcd(e−1,m)−2−gcd(e+1,m)−2) ) if m is even. e∈Inv(m) (2 1 1 Since the term 2 2 (m+gcd(e−1,m)−2) − 2 4 (m+gcd(e−1,m)+gcd(e+1,m)−c) in the formula for chiral embeddings clearly outweighs the corresponding term 1 2 4 (m+gcd(e−1,m)+gcd(e+1,m)−c) in the formula for reflexible embeddings (with c = 3 or 2), this shows that the ratio of reflexible to chiral embeddings tends to zero for large even n. The numbers of regular embeddings of Qn for small values of n (from 3 to 36) are given in Table 1. J Algebr Comb (2011) 33: 215–238 Table 1 Table of numbers of regular embeddings of Qn for small n 235 n Reflexible Chiral Total 3 2 0 2 4 2 0 2 5 2 0 2 6 4 2 6 7 2 0 2 8 8 4 12 9 2 0 2 10 8 12 20 11 2 0 2 12 16 24 40 13 2 0 2 14 16 56 72 15 4 0 4 16 48 144 192 17 2 0 2 18 32 240 272 19 2 0 2 20 64 480 544 21 4 0 4 22 64 992 1056 23 2 0 2 24 192 2304 2496 25 2 0 2 26 128 4032 4160 27 2 0 2 28 256 8064 8320 29 2 0 2 30 320 16960 17280 31 2 0 2 32 640 34688 35328 33 4 0 4 34 512 65280 65792 35 4 0 4 36 1024 130560 131584 7 Genera and other properties To determine the genus of any regular embedding of Qn , all we need to calculate is the face-size, which is the order of the product en σρ of the two generators of G(σ ) = ρ, en σ . If this face-size is s, say, then the genus g and Euler characteristic χ of the map M(σ ) are given by the Euler–Poincaré formula 2 − 2g = χ = |V | − |E| + |F | = 2n − n2n−1 + n2n /s. 236 J Algebr Comb (2011) 33: 215–238 Now let t be the order of σρ in Sn , and let O be the orbit of the point n under the subgroup of Sn generated by σρ. Then an easy calculation gives t  ei , (en σρ)t = |O| i∈O t t and so the order s of en σρ is given by s = t if |O| is even, or s = 2t if |O| is odd. When n is odd, we know that σ = τe (multiplication by e mod n) for some square root e of 1 in Zn , and since ρ is addition by 1 mod n, it is a straightforward exercise to show that ⎧ ⎪ when e = 1, ⎨2n when e = −1, and s = 2t = 2|O| = 4 ⎪ ⎩ 4n gcd(e+1,n) when 1 < e < n − 1 so that the genus g = g(M(σ )) of the map M(σ ) is given by ⎧ ⎪2n−2 (n − 3) + 1 when e = 1,   ⎨ n−3 g M(σ ) = 2 (n − 4) + 1 when e = −1, and ⎪ ⎩ n−3 2 (2n − 4 − gcd(e + 1, n)) + 1 when 1 < e < n − 1. Note that this corrects an error in calculation of both the face-size and the genus in the first concluding remark of [9, Sect. 4] for cases where 1 < e < n − 1. When n is even, the situation is more complicated. Here we let e = 1σ if this is odd, or e = 1σ + n/2 if 1σ is even (and n/2 is odd). Then letting f denote multiplication by e mod m, we know that σ induces the permutation Bi → Bf (i) on the m blocks Bi = {i, i + m}. It is now a straightforward (but longer) exercise to verify that ⎧ 2n when e ≡ 1 mod m and the permutation σ is even, ⎪ ⎪ ⎪ ⎪ ⎪ n when e ≡ 1 mod m and the permutation σ is odd, ⎪ ⎪ ⎪ ⎨8 when e ≡ −1 mod m and 1σ = m − 1, s = 2|O| = 4 when e ≡ −1 mod m and 1σ = n − 1, ⎪ ⎪ ⎪ ⎪ 4n (σρ)i = n for some i, and ⎪ ⎪ ⎪ gcd(e+1,m) when e ≡ ±1 mod m and m ⎪ ⎩ 2n (σρ)i = n for any i. gcd(e+1,m) when e ≡ ±1 mod m and m Hence for n = 2m, the genus g = g(M(σ )) of the map M(σ ) is given by ⎧ 2n−2 (n − 3) + 1 when e ≡ 1 mod m and σ is even, ⎪ ⎪ ⎪ ⎪ n−2 ⎪ 2 (n − 4) + 1 when e ≡ 1 mod m and σ is odd, ⎪ ⎪ ⎪ ⎪ n−4 σ ⎪ ⎪ ⎪2 (3n − 8) + 1 when e ≡ −1 mod m and 1 = m − 1, ⎪ ⎨2n−3 (n − 4) + 1 when e ≡ −1 mod m and 1σ = n − 1, ⎪2n−3 (2n − 4 − gcd(e + 1, m)) + 1 ⎪ ⎪ i ⎪ ⎪ ⎪ when e ≡ ±1 mod m and m(σρ) = n for some i, ⎪ ⎪ ⎪ ⎪ 2n−2 (n − 2 − gcd(e + 1, m)) + 1 ⎪ ⎪ ⎪ i ⎩ when e ≡ ±1 mod m and m(σρ) = n for all i. J Algebr Comb (2011) 33: 215–238 237 It follows that whether n is even or odd, the maximum genus of all orientablyregular embeddings of Qn is 2n−2 (n − 3) + 1 (attained in some cases when e = 1), while the minimum genus is 2n−3 (n − 4) + 1 (attained in some cases when e = −1). Another observation we can make is that if the map M(σ ) is reflexible, then τ σ is not just an involution, but an admissible involution; indeed the map M(τ σ ) is the Petrie dual of M(σ ). On the other hand, if M(σ ) is chiral, then τ σ τ is an admissible involution, and M(τ σ τ ) is the mirror image of M(σ ). Thus orientably-regular embeddings of Qn always come in mated pairs, with each map being the Petrie dual or mirror image of its mate. More generally, we may consider the effect of the ‘hole operators’ considered in [27]. For each j coprime to n, applying the operator Hj to an n-valent map M gives a map Hj (M) with the same underlying graph as M. Here Hj (M(σ )) is M(τj σ τj−1 ), given by the admissible involution τj σ τj−1 (where τj is multiplication by j mod n). Finally, we add the following: Theorem 7.1 All the maps obtained from orientably-regular embeddings of Qn are regular Cayley maps, in the sense that the automorphism group of the map contains a subgroup that acts regularly on vertices. Proof The group of all orientation-preserving automorphisms of the map is the subgroup G = en σ, ρ of the wreath product Z2 ≀ Sn , and so has a natural transitive but imprimitive action on the set {1, 2, . . . , 2n}, with n blocks Bi = {i, i + n} of size 2. The cyclic subgroup Y generated by ρ, which permutes these n blocks in a cycle, is the stabilizer in G of a vertex of the map M(σ ). Now if H is the stabilizer in G of any block, say Bn = {n, 2n}, then H is complementary to Y in G (that is, G = H Y with H ∩ Y = 1), and so H acts regularly on the vertices of M(σ ), which is therefore a regular Cayley map for H .  Acknowledgements The authors acknowledge the use of GAP [11] and M AGMA [2] in analyzing small examples and testing conjectures in the preparation of this paper. The first author’s work was supported by the UI&D Matemática e Aplicações of the University of Aveiro, through the Programa Operacional Ciência, Tecnologia, Inovaçao (POCTI) of the Fundação para a Ciência e a Tecnologia (FCT), and co-financed by the European Community fund FEDER. The second author’s work was supported by the Marsden Fund of New Zealand, project number UOA-721. The third author’s work was supported by the research grants NNSF(10971144) and BNSF(1092010) in China. The fourth author’s work was supported by a Korea Research Foundation grant, funded by the Korean Government (MOEHRD, Basic Research Promotion Fund), number KRF-2008-331-C00049. The fifth author’s work was supported by grant VEGA 1/0722/08 of the Slovak Republic’s Ministry of Education. References 1. Biggs, N.L.: Classification of complete maps on orientable surfaces. Rend. Mat. 4(6), 132–138 (1971) 2. Bosma, W., Cannon, J., Playoust, C.: The M AGMA algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997) 3. Catalano, D.A., Nedela, R.: A characterization of regular embeddings of n-dimensional cubes. Discrete Math. (2010, to appear). doi:10.1016/j.disc.2010.05.010 4. Conder, M.D.E.: Regular maps and hypermaps of Euler characteristic −1 to −200. J. Comb. Theory Ser. B 99, 455–459 (2009). With associated lists of computational data available at the website http://www.math.auckland.ac.nz/~conder/hypermaps.html 238 J Algebr Comb (2011) 33: 215–238 5. Conder, M.D.E., Dobcsányi, P.: Determination of all regular maps of small genus. J. Comb. Theory Ser. B 81, 224–242 (2001) 6. Du, S.F., Jones, G.A., Kwak, J.H., Nedela, R., Škoviera, M.: Regular embeddings of Kn,n where n is a power of 2. I: Metacyclic case. Eur. J. Comb. 28, 1595–1609 (2007) 7. Du, S.F., Jones, G.A., Kwak, J.H., Nedela, R., Škoviera, M.: Regular embeddings of Kn,n where n is a power of 2. II: Non-metacyclic case. Eur. J. Comb. (2010, to appear). doi:10.1016/j.ejc.2010.01.009 8. Du, S.F., Kwak, J.H., Nedela, R.: Regular maps with pq vertices. J. Algebraic Comb. 19, 123–141 (2004) 9. Du, S.F., Kwak, J.H., Nedela, R.: Classification of regular embeddings of hypercubes of odd dimension. Discrete Math. 307, 119–124 (2007) 10. Du, S.F., Kwak, J.H., Nedela, R.: Regular embeddings of complete multipartite graphs. Eur. J. Comb. 26, 505–519 (2005) 11. The GAP Group: GAP—Groups Algorithms and Programming, Version 4.3 (2002). http://www. gap-system.org 12. Gardiner, A., Nedela, R., Širáň, J., Škoviera, M.: Characterization of graphs which underlie regular maps on closed surfaces. J. Lond. Math. Soc. 59, 100–108 (1999) 13. Heffter, L.: Über metacyclic Gruppen und Nachbarconfigurationen. Math. Ann. 50, 261–268 (1898) 14. James, L.D., Jones, G.A.: Regular orientable imbeddings of complete graphs. J. Comb. Theory Ser. B 39, 353–367 (1985) 15. Jones, G.A.: Regular embeddings of complete bipartite graphs: classification and enumeration. Proc. Lond. Math. Soc. (to appear). doi:10.1112/plms/pdp061 16. Jones, G.A., Nedela, R., Škoviera, M.: Regular embeddings of Kn,n where n is an odd prime power. Eur. J. Comb. 28, 1863–1875 (2007) 17. Jones, G.A., Nedela, R., Škoviera, M.: Complete bipartite graphs with a unique regular embedding. J. Comb. Theory Ser. B 98, 241–248 (2008) 18. Jones, G.A., Singerman, D.: Belyi functions, hypermaps, and Galois groups. Bull. Lond. Math. Soc. 28, 561–590 (1996) 19. Kwak, J.H., Kwon, Y.S.: Regular orientable embeddings of complete bipartite graphs. J. Graph Theory 50, 105–122 (2005) 20. Kwak, J.H., Kwon, Y.S.: Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs. Discrete Math. 308, 2156–2166 (2008) 21. Kwon, Y.S.: New regular embeddings of n-cubes Qn . J. Graph Theory 46, 297–312 (2004) 22. Kwon, Y.S., Nedela, R.: Non-existence of nonorientable regular embeddings of n-dimensional cubes. Discrete Math. 307, 511–516 (2007) 23. Nedela, R., Škoviera, M.: Regular maps of canonical double coverings of graphs. J. Comb. Theory Ser. B 67, 249–277 (1996) 24. Nedela, R., Škoviera, M.: Regular maps from voltage assignments and exponent groups. Eur. J. Comb. 18, 807–823 (1997) 25. Nedela, R., Škoviera, M., Zlatoš, A.: Bipartite maps, Petrie duality and exponent groups. Atti Semin. Mat. Fis. Univ. Modena 49 (Suppl.), 109–133 (2001). Dedicated to the memory of Professor M. Pezzana (in Italian) 26. Nedela, R., Škoviera, M., Zlatoš, A.: Regular embeddings of complete bipartite graphs. Discrete Math. 258, 379–381 (2002) 27. Wilson, S.E.: Operators over regular maps. Pac. J. Math. 81, 559–568 (1979) 28. Xu, J.: A classification of regular embeddings of hypercubes Q2m with m odd. Sci. China Ser. A, Math. 50, 1673–1679 (2007)