J Math Chem https://doi.org/10.1007/s10910-018-0900-y ORIGINAL PAPER

P SL (2, 7) and carbon allotrope D168 Schwarzite Qaiser Mushtaq1 · Nighat Mumtaz2

Received: 5 November 2017 / Accepted: 20 March 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract We investigate actions of the modular group P S L(2, Z) on the projective line over finite fields P L(F7n ) and find interesting relation between the coset diagram of orbits and the carbon allotrope with negative curvature D168 Shewarzite. We also highlighted some topological aspects of these diagrams. Keywords Projective special linear groups · Heptakisoctahedral group · Coset diagrams · Genus · Euler’s characteristics Mathematics Subject Classification Primary 05C38 · 15A15; Secondary 05A15 · 15A18

1 Introduction The point groups in chemistry portray the spatial symmetry of molecules [11,12]. In this context the groups of the regular polyhedra are specifically noteworthy because of their point symmetry. It is discussed in [16] that these regular polyhedral groups are subgroups of larger permutation groups, which themselves are subgroups of the corresponding symmetric groups Sn . Of specific pertinence to chemists in [15] is that these groups may be utilized to depict carbon allotrope structures with negative curvature built from hexagons and heptagons of sp2-hybridized carbon atoms [7,15,

B

Nighat Mumtaz [email protected] Qaiser Mushtaq [email protected]

1

The Islamia University of Bahawalpur, Bahawalpur, Pakistan

2

Quaid-i-Azam University, Islamabad, Pakistan

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Fig. 1 Klein-Graph

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25]. Coset diagrams for the extended modular group P G L(2, Z), initiated by Higman (see [4,13,21]) have turned out to be very useful in understanding spatial symmetry of Fullerene molecules. P S L(2, p) contains a special subset of groups for p = 5, 7, 11, in perspective of their specific structure of permutations. In three dimensional space the pollakispolyhedral groups can be viewed as multiples of regular polyhedral symmetry groups [7]. In this paper we are interested in P S L(2, 7) having order 168 and is 7 O the heptakisoctahedral group. It has a subgroup of index 7 which is the octahedral group “O”. The rotational symmetry of an idealization of the “plumber’s nightmare” is P S L(2, 7), which is a representation for carbon allotropes “Schwarzite” [20]. Geometrical models, for the group P S L(2, 7) or heptakisoctahedral group of order 168 depict its transitive permutations on sets of 7 and 8 objects. A set of seven objects permuted transitively by the group P S L(2, 7) can be acquired when an equilateral triangle and a inscribed circle form the seven-point-seven-line geometry presented in D3 symmetry [15] (Fig. 1). The seven collineations (AE B, AGC, B FC, B DG, AD F, E F G, C D E) preserved by the permutations of the seven vertex labels form the group P S L(2, 7). Note here that in this presentation the six straight lines making the three altitudes and the three edges of the triangle and the inscribed circle are treated on an equal basis. Eight objects permuted transitively by the heptakisoctahedral group are the vertex labels of a cuboid of D2 point group symmetry which give a set of 168 nonsuperimposable cuboids, form the group P S L(2, 7). In analogy to the connection between the tetrahedral and icosahedral group, the octahedral rotation group O can be obtained from the heptakisoctahedral group or P S L(2, 7) by erasing all seven-fold symmetry elements [7]. The regular genus-3 Klein map group is another representation of P S L(2, 7) . Its high symmetry in association with the theory of multivalued functions is studied in [16]. This map shows the transitivity on a 7-set, when sevenfold symmetry elements removed, seven octahedral structures are obtained which contain eight vertices. The

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Fig. 2 Seven Point-Seven line geometry

relation of carbon structures with negative-curvature and this group can be found in [1,6]. The group P S L(2, 7) is important to analyze the permutational symmetry of D168 Schwarzite. In fact the prototypical role of D168 in Schwarzite series and C60 in fullerene series relate that the carbon atoms in their structure and the order of corresponding transitive permutational group are same. The structure of D168 is derived from a unit cell of 24 heptagons embedded in a surface of genus 3. These 24 heptagons have 56 vertices as every heptagon contains 7 vertices and three heptagons are connected with each other with one vertex. Infinite minimal surfaces with minimal Gaussian curvature and surfaces with genus 3 are discussed in [15]. This Fig. 2 (discussed by Klein in [17]) portrays an open network of full heptagons or their portions that can be modified into a negative curvature of genus 3. The unit cell with 24 heptagons, 84 edges and 56 vertices has an Euler’s characteristic which corresponds to the genus 3. A carbon allotrope with such type of structure is known as D56 protoschwarzite. This carbon allotrope structure leads to D168 structure, which is discussed in details in [15]. A Hurwitz group [5] is any finite group generated by elements r and s of order 2 and 3 and order of r s is 7. Equivalently, a Hurwitz group is any finite nontrivial quotient of (2, 3, 7) with presentation < r, s : r 2 = s 3 = (r s)7 = 1 > . The significance of the latter group (and its quotients) is best explained by referring to some aspects of the theory of Fuchsian groups, hyperbolic geometry, Riemann surfaces, and triangle groups. Details are available in books by Beardon [2] and Jones and Singerman [14]. The group P S L(2, 7) is the group of 2 × 2 matrices entries from the field Z7 with determinant 1. It is known as the smallest Hurwitz group of order 168, representing

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the group of Klein’s quartic curve x 3 y + yz + z 3 x = 0. It can be obtained by the linear fractional transformations ω → −1/ω and ω → (ω − 1) /ω of the projective line over the field Z7 . These transformations having orders 2 and 3, respectively, with their product being ω → ω + 1 having order 7. The group P S L(2, 7) is known to have the presentation < r, s : r 2 = s 3 = (r s)7 = [r, s]4 = 1 >, where [r, s] denotes the commutator of r and s. Similar presentations are considered by different authors, with an extra relator inserted (See [8,18,19,24]). The P S L(2, Z) is group of all linear fractional transformations of type ω → (cω + d) / (lω + m) with cm − ld = 1 and c, d, l, m ∈ Z. The elements μ : ω → −1/ω and ν : ω → (ω − 1) /ω generate this group and μ, ν : μ2 = ν 3 = 1 is its finite presentation. Let q = p n be a power of a prime p for some positive integer n. Then by P L(Fq ) we mean the projective line over the finite field Fq containing the elements of Fq , together with an additional point ∞. In particular, if n = 1 then P L(Fq ) is P L(F p ) = {1, 2, 3, . . . , p − 1} ∪ {∞} . The group P S L(2, 7) is obtainable when P S L(2, Z) acts on P L (F7n ) , where P L (F7n ) = F7n ∪ {∞}. We are interested to investigate the action of P S L(2, Z) on P L (F7n ) and find some relevance in the orbits of the coset diagram of the group obtained and the carbon D168 Schwarzite. A coset diagram for a finitely generated group is a graph whose each vertex represents a right coset of a stabilizer subgroup with finite index. If vgi = u, then a gi − edge of “colour i” joins the vertices representing v and u, and is directed from vertex v to the vertex u . v → vgi = u In case when vgi = v, then v−vertex is connected to itself by a gi − loop or we can say that v is a fixed point of gi . We draw coset diagrams for the action of P G L(2, Z) or P S L(2, Z) on P L (F7n ) and consider them as the diagrammatic analog of D168 Schwarzite. Since P S L(2, Z) is generated by two elements μ and ν , the three cycles of ν are denoted by small triangles whose vertices are permuted counter-clockwise by ν and involution μ, is denoted by an edge. Heavy dots represent the fixed points of μ and ν. As discussed in [22], the third generator t : ω → 1/ω belonging to P G L(2, Z) represent the reflection along the vertical axis in the diagram.

2 Action of G on P L(F7n ) In this section we discuss the action of G = P S L(2, Z) on P L(F7n ) for different values of n, where n ∈ N. We draw coset diagrams to investigate the properties of this action and the orbits of the group thus obtained. We follow [23] to prove the theorems to keep the paper self-contained.

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Fig. 3 γ1

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2.1 Action of G on P L(F7 ) ¯ as a group generated by μ¯ and ν¯ , where μ¯ and ν¯ are the permutation Consider G representations of μ and ν after the action of G on P L(F7n ) for n ∈ N. Taking n = 1, the action of G on P L(F7 ) gives μ¯ = (0 ∞) (1 6) (2 3) (4 5) and ν¯ = (1 0 ∞) (2 4 6) (3) (5) .This yields the following coset diagram γ1 which can be graphically represented as γ1 (Fig. 3). This diagram is a representation of the well known simple group of order 168 [9]. 2.2 Action of G on P L(F72 ) Now the group G acting on P L(F72 ). An irreducible polynomial of degree 2 in F72 is t 2 + 2t + 3 . The elements of F49 are of the form t0 + t1 ζ, where ti ∈ Z7 , for i = 0, 1. Let ζ be the primitive root of G F (49) satisfying ζ 2 = 5ζ + 4. When G acts on P L(F72 ), μ and ν have the following permutation representation        μ¯ = (0 ∞) 1 ζ 24 ζ ζ 23 ζ 2 ζ 22 ζ 3 ζ 21 ζ 4 ζ 20 ζ 5 ζ 19         ζ 6 ζ 18 ζ 7 ζ 17 ζ 8 ζ 16 ζ 9 ζ 15 ζ 10 ζ 14 ζ 11 ζ 13 ζ 12 ζ 12        ζ 25 ζ 47 ζ 26 ζ 46 ζ 27 ζ 45 ζ 28 ζ 44 ζ 29 ζ 43 ζ 29 ζ 42        ζ 30 ζ 41 ζ 31 ζ 40 ζ 32 ζ 39 ζ 33 ζ 38 ζ 34 ζ 37 ζ 35 ζ 36 and       ν¯ = (1 0 ∞) ζ ζ 2 ζ 21 ζ 3 ζ 7 ζ 14 ζ 4 ζ 31 ζ 37 ζ 5 ζ 13 ζ 6 ζ 8       ζ 9 ζ 30 ζ 33 ζ 10 ζ 36 ζ 26 ζ 11 ζ 17 ζ 44 ζ 12 ζ 38 ζ 22 ζ 40       ζ 16 ζ 32 ζ 16 ζ 15 ζ 18 ζ 39 (ζ 19 ζ 28 ζ 25 ) ζ 20 ζ 29 ζ 23 ζ 27 ζ 46 ζ 47    ζ 35 ζ 43 ζ 42 ζ 34 ζ 41 ζ 45

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which yield the following two orbits γ1 (Fig. 3) and γ2 (Fig. 4). γ2 can be graphically represented as: This coset diagram represents a group of order 168 [9] and consists of two orbits γ1 and γ2 . 2.3 Action of G on P L(F73 ) For n = 3, let β 3 +6β +2 be the irreducible polynomial in F73 . The field has elements of type β0 + β1 ζ + β2 ζ 2 , where βi ∈ Z7 , for i = 0, 1, 2. Let ζ be the 342th primitive root of unity satisfying ζ 3 = ζ + 5 of F343 . When G acts on P L(F73 ), we have orbits γ1 and γ3 (Fig. 5) graphically represented as: This coset diagram represents a group of order 168 [9] and consists of two orbits γ1 and two copies of γ3 . The orbit γ3 with 24 heptagons has 56 triangles where each triangle is shared by three heptagons, e = (24)(7) /2 + 56 (3) = 252 edges, v = 168 vertices and f = 24 + 56 = 80 faces. Thus has Euler’s characteristics (168 − 252 + 80) = −4 which corresponds to genus 3. The diagrammatic structure of this orbit is similar to the structure of D168 Schwazite as both have same genus. Also total number of carbon atoms in D168 schwarzite structure and the order of permutational group obtained are same. In this paper we are considering the action of G on P L(F7n ) for n ≤ 3 because the orbits of the action for n ≥ 4 contain no new coset diagrams for the orbits other than γ1 , γ2 and γ3 in the coset diagram. Therefore the action of G on P L(F7n ) for n ≥ 4 is not discussed. Theorem 2.1 If G acts on P L(F7n ), n ∈ N ¯ =< μ, G ¯ ν¯ : (μ) ¯ 2 = (¯ν )3 = (μ¯ ¯ ν )7 = [μ¯ ν¯ ]4 = 1 >∼ = P S L (2, 7) .

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Proof Indeed the actions considered are homomorphisms from P S L (2, 7) to Sym(m), for m = 8,42,168, whose images are transitive subgroups. Obviously these images are isomorphic to P S L (2, 7) , since this group is simple.  Existence of fixed points of μ¯ and ν¯ in these coset diagrams play important role which will be evident in the subsequent discussion. Theorem 2.2 If G acts on P L(F7n ), then (1) Fixed points of μ¯ exist only for even n. (2) Fixed points of ν¯ exist for all n. Proof (1) When n is even, 7n + 1 are the total number of elements in P L(F7n ). As we have 7n ≡ 1 (mod4) and the permutation μ¯ composed of two cycles leaving one element which becomes a fixed point of μ. ¯ (2) (ω) ν = (ω − 1) /ω implies (ω − 1) /ω = ω, that is ω2 − ω + 1 = 0. So ω ≡ 3, 5 (mod7) are fixed points of ν¯ exist for all n.  Remark 2.3 The action of G on P L(F7n ) gives three types of orbits γ1 ,γ2 and γ3 (Figs. 3, 4, 5). The orbit γ1 consist of 8 elements. ζ (q−1)/6 and ζ 5(q−1)/6 are fixed points of ν¯ in γ1 where q = 7n . All coset diagrams for this action contain γ1 for all n. ζ (q−1)/4 and ζ 3(q−1)/4 are fixed points of μ¯ which lie in the orbit γ2 consisting of 42 elements. This orbit exists in coset diagram only for even n. The third orbit γ3 consist of 168 vertices but it does not contain any fixed points of μ¯ or ν¯ . It exists in coset diagram always in the form of symmetric pairs for all n  3. Remark 2.4 Let < ζ : ζ 7

n −1

= 1 > be the cyclic subgroup of F7n . Then,

(i) the fixed points of μ¯ are ζ (7

n −1)/4

and ζ 3(7

n −1)/4

,

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(ii) the fixed points are ν¯ are ζ (7 −1)/6 and ζ 5(7 −1)/6 , and n (iii) 0, 1, 2, 3, 4, 5, 6 and ∞ are the vertices of γ1 , where 2 = ζ (7 −1)/3 , 4 = n n ζ 2(7 −1)/3 and 6 = ζ (7 −1)/2 . n

n

Theorem 2.5 If G acts on P L(F7n ), then    n ¯  = 1 + (7 +1)−8 if n is odd, (i)  Or b P L(F7n ) G 168    n ¯  = 2 + (7 +1)−50 if n is even. (ii)  Or b P L(F7n ) G 168 Proof By Remark 2.3, when n is odd, then the orbit γ1 composed of 8 vertices exists ¯ is isomorfor all n. So (7n + 1) − 8 elements of P L(F7n ) are left. By Theorem 2.1 , G phic to P S L (2, 7) containing elements of orders 2, 3, 4, 7 and the identity element. Theorem 2.2 shows that for odd n, there is no fixed point of μ. ¯ So there are 21 elements of order 2 which do not fix any element of P S L (2, 7) . Also, when n is odd, 7n ≡ 3 (mod4). Therefore there are 42 elements of order 4 which do not fix any element. By Theorem 2.1, fixed points of ν¯ exist for all n, therefore there are 56 elements of order 3 fixing 2 elements. Moreover, 7n + 1 ≡ 1 (mod7) implies that there are 24 + 24 = 48 elements of order 7 which fix one element because P S L (2, 7) contains two conjugacy classes of order 7. All (7n + 1) elements of P L(F7n ) are fixed by identity element. By Frobenius–Burnside lemma [10], the total number of orbits is ⎛ ⎞      1   Or b P L(F n ) G ¯ =  ⎝  Fi x P L(F7n ) (g)⎠ 7 G ¯ ¯ g∈G

  1  = 21 × 0 + 42 × 0 + 56 × 2 + 48 × 1 + 1 × 7n + 1 168 (7n + 1) − 8 = 1+ 168 By Remark 2.3, when n is even, γ1 containing 8 and γ2 containing 42 vertices, are two orbits. Only when n is even, γ2 exists in coset diagram. So (7n + 1) − 50 elements of P L(F7n ) are left. By Theorem 2.2 when n is even, fixed points of μ¯ exist so 21 elements of order 2 fix two elements. When n is even we have 7n ≡ 1 (mod4). Therefore 42 elements of order 4 are fixing 2 elements. Fixed points of ν¯ exist for all n so 56 elements of order 3 are fixing two elements. In addition 7n + 1 ≡ 1 (mod7) , so 48 elements of order 7 are fixing one element and all (7n + 1) elements are fixed by identity element. By Frobenius–Burnside lemma [10], the total number of orbits is ⎛ ⎞         Or b P L(F n ) G ¯  = 1 ⎝  Fi x P L(F7n ) (g)⎠ 7 G ¯ ¯ g∈G

  1  21 × 2 + 42 × 2 + 56 × 2 + 48 × 1 + 1 × 7n + 1 168 (7n + 1) − 50 = 2+ 168

=

 This leads us to the following corollary.

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Corollary 2.6 The action of G on P L(F7n ) is intransitive. number of orbits, including one orbit Remark 2.7 (1) If n is odd, we have 1+ (7 +1)−8 168 n number γ1 containing 8 vertices. Leftover elements are evenly divided into (7 +1)−8 168 of orbits. All of these orbits are copies of γ3 consisting of 168 vertices. n number of orbits. One of these orbits is γ1 (2) If n is even, we have 2 + (7 +1)−50 168 containing 8 vertices and the other is γ2 containing 42 vertices. Remaining elements n are evenly divided into (7 +1)−8 number of orbits. All these orbits are copies of γ3 168 containing 168 vertices. n

Theorem 2.8 For any prime p, p n = { p + l. p ( p − 1) + s.

  p p 2 −1 2

}.

Proof For n = 1, p n = p and for n = 2, we have p 2 = p + p ( p − 1) .  p p 2 −1

Suppose for n = k, it is true, that is p k = { p + l. p ( p − 1) + s. 2 } where l = 0 if n is odd, l = 1 if n is even and s = 0 for n < 3. p p 2 −1 }. p. which implies For n = k + 1, consider p k . p = { p + l. p ( p − 1) + s. 2   p 2 p 2 −1

that p k+1 = { p 2 + l. p 2 ( p − 1) + s. } = p + p ( p − 1) + l. p ( p − 1) − 2      2  p 2 p 2 −1 p p 2 −1 l. p p − 1 + s. = p + + l) { p p − 1)} + (sp − 2l) = p+ (1 ( 2  2    p p 2 −1 . l . p ( p − 1) + s 2 Therefore it is true for all values of n.



3 Conclusion The group P S L(2, 7) is an important group of order 168 and has many applications in carbon chemistry. It is useful to understand and analyze the structure of graphite and fullerenes having surface of negative curvature due to its link with polymeric carbon allotropes having unusually low density. We analyzed that the coset diagrams for the action of P G L(2, Z) or P S L(2, Z) on P L (F7n ) , are a diagrammatic view of D168 n Schwarzite. The total number of orbits that exist in coset diagram are 1 + (7 +1)−8 168 n if n is odd and 2 + (7 +1)−50 if n is even. The orbits of the coset diagram are closely 168 related to the structure of D168 Schwarzite. The transitive action of G on a set of 7 elements for n = 1 gives us an orbit γ1 having 8 vertices and also has octahedral “O” symmetry. It is 7 O heptakisoctahedral group [3]. For n = 2, G acts on P L (F49 ) intransitively obtaining two orbits γ1 and γ2 containing 8 and 42 elements respectively and representing heptakisoctahedral group. When G acts on P L (F7n ) for n  3, we obtain orbits γ1 , γ2 and copies of γ3 . The orbit γ3 and D168 Shwarzite are topologically same as both have genus 3. The total number of carbon atoms in D168 schwarzite structure and the order of permutational group obtained are also same.

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