Classifying semisymmetric cubic graphs of order 20p

A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. In this paper we classify all connected cubic semisymmetric graphs of order 20p , p prime.


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In this paper all graphs are finite, undirected and simple, i.e. without loops and multiple edges. A graph is 9 called semisymmetric if it is regular and edge-transitive but not vertex-transitive. 10 The class of semisymmetric graphs was first studied by Folkman [9], who found several infinite families 11 of such graphs and posed eight open problems. 12 An interesting research problem is to classify connected cubic semisymmetric graphs of various types of 13 orders. In [9], Folkman proved that there are no semisymmetric graphs of order 2p or 2p 2 for any prime p . 14 The classification of semisymmetric graphs of order 2pq , where p and q are distinct primes, was given in [7]. 15 For prime p, cubic semisymmetric graphs of order 2p 3 were investigated in [17], in which the authors 16 proved that there is no connected cubic semisymmetric graph of order 2p 3 for any prime p ̸ = 3 and that for 17 p = 3 the only such graph is the Gray graph. 18 Also connected cubic semisymmetric graphs of orders 4p 3 , 6p 2 , 6p 3 , 8p 2 , 8p 3 , 10p 3 , 18p n (n ≥ 1) have 19 been classified in [1,2,8,11,13,21]. 20 In this paper we investigate connected cubic semisymmetric graphs of order 20p for all primes p . Note 21 that for orders like 4p , 6p, 10p and 14p which are of the form 2qp for some fixed prime q , the problem of 22 classifying such graphs follows from the general result of [7]. 23 We prove that if Γ is a connected cubic semisymmetric graph of order 20p , p prime, then p = 11 24 and Γ is isomorphic to a known graph. We go beyond however and prove that there is no connected cubic 25 G -semisymmetric graph of order 20p , for any prime p ̸ = 2, 11. This will put us near the classification of all 26 connected cubic G -semisymmetric graphs of order 20p : if there is any such graph, then its order must be either

Preliminaries 2
In this paper the symmetric and alternating groups of degree n , the dihedral group of order 2n and the cyclic 3 group of order n are respectively denoted by S n , A n , D 2n , Z n . If G is a group and H ≤ G , then Aut (G), G ′ , 4 Z(G) , C G (H) and N G (H) denote respectively the group of automorphisms of G , the commutator subgroup 5 of G , the center of G , the centralizer and the normalizer of H in G . We also write H ¢ c G to denote H is a 6 characteristic subgroup of G . If H ¢ c K ¢ G , then H ¢ G . For a prime p dividing the order of a finite group 7 G , O p (G) will denote the largest normal p -subgroup of G . It is easy to verify that O p (G) ¢ c G . 8 For a group G and a nonempty set Ω, an action of G on Ω is a function (g, ω) → g.ω from G × Ω to 9 Ω , where 1.ω = ω and g.(h.ω) = (gh).ω , for every g, h ∈ G and every ω ∈ Ω . We write gω instead of g.ω , if 10 there is no fear of ambiguity. For ω ∈ Ω, the stabilizer of ω in G is defined as G ω = {g ∈ G : gω = ω}. The 11 action is called semiregular if the stabilizer of each element in Ω is trivial; it is called regular if it is semiregular 12 and transitive.

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For any two groups G and H and any homomorphism φ : H → Aut(G) the external semidirect product 14 G⋊ φ H is defined as the group whose underlying set is the cartesian product G×H and whose binary operation to be the internal semidirect product of N and K . These two concepts are in fact equivalent in the sense that 18 there is some homomorphism φ : The dihedral group D 2n is defined as denoted by V (Γ) , E(Γ), Arc (Γ) and Aut(Γ) . If Γ is a graph and N ¢Aut (Γ) , then Γ N will denote a simple 27 undirected graph whose vertices are the orbits of N in its action on V (Γ) , and where two vertices N u and N v 28 are adjacent if and only if u ∼ nv in Γ, for some n ∈ N .

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Let Γ c and Γ be two graphs. Then Γ c is said to be a covering graph for Γ if there is a surjection is a one to one correspondence. f is called a covering projection. Clearly, if Γ is bipartite, 32 then so is Γ c . For each u ∈ V (Γ), the fibre on u is defined as f ib u = f −1 (u) . The following important set is 33 a subgroup of Aut (Γ c ) and is called the group of covering transformations for f : It is known that K = CT (f ) acts semiregularly on each fibre [14]. If this action is regular, then Γ c is said to 36 be a regular K -cover of Γ. 37 Let X ≤Aut (Γ). Then Γ is said to be X -vertex transitive, X -edge transitive or X -arc transitive if 38 X acts transitively on V (Γ), E(Γ) or Arc(Γ) respectively. The graph Γ is called X -semisymmetric if it is 39 regular and X -edge transitive but not X -vertex transitive. Also Γ is called X -symmetric if it is X -vertex 1 transitive and X -arc transitive. For X =Aut (Γ), we omit X and simply talk about Γ being edge transitive, 2 vertex transitive, symmetric or semisymmetric. As an example, Γ = K 3,3 , the complete bipartite graph on 6 3 vertices, is not semisymmetric but it is X -semisymmetric for some X ≤ Aut(Γ) .

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An X -edge transitive but not X -vertex transitive graph is necessarily bipartite, where the two partites 5 are the orbits of the action of X on V (Γ). If Γ is regular, then the two partite sets have equal cardinality. So 6 an X -semisymmetric graph is bipartite such that X is transitive on each partite but X carries no vertex from 7 one partite set to the other.

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According to [5], if there is a unique known cubic semisymmetric graph of order n, then it is denoted by  Any minimal normal subgroup of a finite group, is the internal direct product of isomorphic copies of a 12 simple group.

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A finite group G is called a K n -group if its order has exactly n distinct prime divisors, where n ∈ N .

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Theorem 2.1 (i) If G is a simple K 3 -group, then G is isomorphic to one of the following groups:

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(2) L 2 (r) where r is a prime, r 2 − 1 = 2 a · 3 b · s, s > 3 is a prime, a, b ∈ N ;  An immediate consequence of the following theorem of Burnside is that the order of every nonabelian 28 simple group is divisible by at least 3 distinct primes.

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In the following theorem, the inverse of a pair (a, b), is meant to be (b, a). Also for each i , and D i are certain groups of order i with known structures. We will not need their structures. vertex is of the form 2 r · 3 for some 0 ≤ r ≤ 7. More precisely, if {u, v} is any edge of Γ, then the pair 2 (X u , X v ) can only be one of the following fifteen pairs or their inverses: Proposition 2. 5 [17] Let Γ be a connected cubic X -semisymmetric graph for some X ≤ Aut(Γ); then either 6 Γ ≃ K 3,3 , the complete bipartite graph on 6 vertices, or X acts faithfully on each of the bipartition sets of Γ.

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Our goal in this paper is to fully classify connected cubic semisymmetric graphs of order 20p . We also derive a 30 very restrictive necessary condition for the existence of connected cubic G -semisymmetric graphs of order 20p . 31 We prove the following important result. Part (i) is a full classification whereas part (ii) is only a necessary 32 condition.

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Theorem 3.1 Let p be a prime. 1 (i) If Γ is a connected cubic semisymmetric graph of order 20p , then p = 11 and Γ ≃ S220 .
To prove the main theorem, we need some lemmas.

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As for sub-item (3), let L 2 (2 m ) be a group of order 2 i · 3 · 5 · p; then where m, 2 m − 1 and 2 m +1 3 are all primes according to Theorem 2.1. This equation has no answer as neither 12 2 m − 1 nor 2 m +1 3 could be equal to 5. Finally consider groups L 2 (r) in sub-item (2) . If for odd prime r and 13 for prime s > 3, we have  divides |N | . Since the order of every simple K 3 -group is divisible by 3 , N must be a simple K 4 -group whose 3 order is of the form 2 i · 3 · 5 · p . According to Lemma 3.2, N ≃ L 2 (2 4 ), L 2 (11) or L 2 (31) corresponding to 4 p = 17 , 11 and 31 respectively. But these cases are ruled out in the statement of the Lemma. If N ≃ Z 2 , then | G N | = 2 r · 3 · 5 · p and |U N | = |W N | = 5p . If M N is unsolvable, then it must be a simple 10 K 4 -group whose order is of the form 2 i · 3 · 5 · p. It follows from Lemma 3.2, that p = 17 , 11 or 31 which are also call H of type D if there is at least one element in H whose second component is not in A 5 . Also for 12 any x ∈ Z p and any g, h ∈ S 5 we define two subsets R x,g,h , S x,g,h ⊂ Z p ⋊ φ S 5 as follows: As we will see later, these two subsets are sometimes subgroups of Z p ⋊ φ S 5 .

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The group D 12 = ⟨ a, b|a 6 = b 2 = 1, b −1 ab = a −1 ⟩ has exactly three Sylow 2 -subgroups, all isomorphic 18 to Z 2 × Z 2 , which are listed below: to see that for any positive integer n if g ∈ A 5 , then (x, g) n = (x n , g n ) for all x ∈ Z p , and if g / ∈ A 5 , then and hence H is isomorphic to a subgroup of A 5 . As A 5 has no element of order 6 , H cannot be isomorphic 19 Therefore m = n = 6 and so |H| = 6. Since A 4 does not have a subgroup of order 6 , we conclude that 20 H ≃ D 12 .

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We now proceed to prove part (ii). So let D 12 ≃ H ≤ Z p ⋊ φ S 5 . According to part (i) H is of type D. 22 We continue to use the notations invented in the proof of part (i) . The group D 12 has only two subgroups of 23 order 6 , namely Z 6 and S 3 . Since H ≤ H and |H| = 6 , we have H ≃ Z 6 or S 3 . Since H ≃ H 1 ≤ A 5 and A 5 24 does not have elements of order 6, it follows that H cannot be isomorphic to Z 6 and hence H ≃ S 3 . Also as   1), (x, g), (1, g 2 ), (x, g 3 ), (1, g 4 ), (x, g 5 ), 1 (x, g ′ ), (1, g ′ g), (x, g ′ g 2 ), (1, g ′ g 3 ), (x, g ′ g 4 ), (1, g ′ g 5 )} = S x,g,g ′ . x ∈ Z p , some g, k / ∈ A 5 and some h ∈  On the other hand if V ≃ D 12 , then according to Lemma 3.6, either V = R y,g ′ ,h ′ or V = S y,g ′ ,k ′ for 31 some y ∈ Z p , some g ′ , k ′ / ∈ A 5 and some h ′ ∈ A 5 . Again all the sylow 2 -subgroups of V are known. The first  bipartite cubic symmetric graph of order n or it is a cubic semisymmetric graph of order n. 8 We now set off to prove part (ii) of Theorem 3.1. For p = 3, 5, 7, 17, 31 there is no connected cubic 9 semisymmetric graph of order 20p according to [5]. Also for p = 5, 7, 17 no connected cubic symmetric graph 10 of order 20p exists according to [6]. As for p = 3, 31 , according to [6] there exists only one connected cubic 11 symmetric graph of order 20p which is not bipartite. Therefore we conclude that for p = 3, 5, 7, 17, 31 there is 12 no connected cubic G -semisymmetric graph of order 20p .

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Now let p > 11 be a prime such that p ̸ = 17, 31. Suppose on the contrary that Γ is a connected cubic symmetric graph or a cubic semisymmetric graph of order 20 . By [5] there is no semisymmetric cubic 20 graph of order 20 and by [6] there is only one bipartite symmetric cubic graph of order 20 , namely F20B .

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Therefore Γ M ≃ F20B. 22 The automorphism group of F20B has 240 elements ([6]) and G M is isomorphic to a subgroup of only ones whose orders are of the form 2 i · 3 · 5 for 1 ≤ i ≤ 3 , are T 7 ≃ A 5 of order 60 , and T 11 , T 12 and 29 T 13 ≃ S 5 of order 120.

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First note that G M ≃ T 7 is not possible according to Lemma 3.5. Next, we argue that G M could not 31 be isomorphic to T 11 or T 12. Again, using GAP one finds out that this group is nonabelian of order 12 which has the following group as a 9 normal subgroup:  Also Γ ≃ C(G; G u , G v ) by Proposition 2.9. Now it follows from part (ii) of Proposition 2.8, that 19 G = ⟨G u , G v ⟩ and from part (i) of the same Proposition that |G u ∩ G v | = 4 ; i.e. G u ∩ G v is a common Sylow 20 2 -subgroup of G u and G v . But the existence of G u and G v with all these properties contradicts Lemma 3.7.

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Since every case for G M is contradictory, part (ii) follows.

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Next, we turn to part (i) of Theorem 3.1. For p ̸ = 2, 11 there is no connected cubic semisymmetric 23 graph of order 20p according to part (ii). Also there is no such graph of order 20 × 2 according to [5] and by 24 the same reference, there is only one connected cubic semisymmetric graph of order 20 × 11, namely S220.

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Acknowledgment 26 The authors would like to thank the anonymous referees for their helpful comments. The second author was 27 supported in part by a grant from the IMU-CDC.