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Outline

Title
Abstract
Figures
Computational Procedures
Low Index Subgroups Methods
Introduction to the Concepts
Uniqueness Result
The Permutation Method of Kwak and Kwon
Classification of Doubly Transitive Designs
Introduction
Non-Planar Case
References

Symmetrical covers, decompositions and factorisations of graphs

Profile image of Cheryl PraegerCheryl Praeger

2008

Abstract

ABSTRACT: This paper introduces three new types of combinatorial structures associated with group actions, namely symmetrical covers, symmetrical decompositions, and symmetrical factorisations of graphs. Thesestructures are related to and generalise various combinatorial objects, such as 2-designs, regular maps, near-polygonal graphs, and linear spaces. General theory is developed for each of these structures, pertinent examples and constructions are given, and a number of open research problems are posed.

Figures (31)
About the editors
About the editors
By further theory of symmetric cubic graphs (developed by Tutte, Goldschmidt, et al), it is now known that if X is a 5-arc-regular cubic graph, then AutX is a homomorphic image of the finitely-presented group
By further theory of symmetric cubic graphs (developed by Tutte, Goldschmidt, et al), it is now known that if X is a 5-arc-regular cubic graph, then AutX is a homomorphic image of the finitely-presented group
Figure 2. Chirality in terms of normal subgroups of A°.
Figure 2. Chirality in terms of normal subgroups of A°.
‘igure 3. Partial illustration of a 3-polytope.
‘igure 3. Partial illustration of a 3-polytope.
Figure 6. Composition of coset graphs for the ordinary (2, 3, 7) triangle group.
Figure 6. Composition of coset graphs for the ordinary (2, 3, 7) triangle group.
Figure 7. An extended coset table. Two drawbacks of the (standard) low index subgroups algorithm are the fact that the finite quotients it produces can have large order but small minimal degree (as permutation groups), and K is the core of H (the intersection of all conjugates of H) in G, as a subgroup of Sp. These images can often be used as the ‘ building blocks’ for the construction of larger images (as in Section 3), or produce interesting examples in their own right. For instance, the first known examples of arc-transitive cubic graphs admitting no edge-reversing automorphisms of order 2, and first known 5-arc-transitive cubic grap h having no s-arc-regular group of automorphisms for s <5, were found in this way (see [18]). The same approach was used to help construct infinite family of 4-arc-transitive connected finite cubic graphs of girth 12, and then (unexpectedly) to a new symmetric presentation for the specia. linear group SL(3, Z); see [7]. Similarly, it enabled the construction of a infinite family of vertex-transitive but non-Cayley finite connected 4-valent graphs with arbitrarily large vertex-stabilizers in their automorphism groups [23], the first known example of a finite half-arc-transitive (vertex- and edge-transitive but not arc-transitive) 4-valent finite graph with non-abelian vertex-stabilizer [20], and the first known examples of finite chiral polytopes of rank 5 [15]. Tn dAravshacke af the (etandard) Inv, indeyvy aenhornimne alanrithm are the fact that the finite
Figure 7. An extended coset table. Two drawbacks of the (standard) low index subgroups algorithm are the fact that the finite quotients it produces can have large order but small minimal degree (as permutation groups), and K is the core of H (the intersection of all conjugates of H) in G, as a subgroup of Sp. These images can often be used as the ‘ building blocks’ for the construction of larger images (as in Section 3), or produce interesting examples in their own right. For instance, the first known examples of arc-transitive cubic graphs admitting no edge-reversing automorphisms of order 2, and first known 5-arc-transitive cubic grap h having no s-arc-regular group of automorphisms for s <5, were found in this way (see [18]). The same approach was used to help construct infinite family of 4-arc-transitive connected finite cubic graphs of girth 12, and then (unexpectedly) to a new symmetric presentation for the specia. linear group SL(3, Z); see [7]. Similarly, it enabled the construction of a infinite family of vertex-transitive but non-Cayley finite connected 4-valent graphs with arbitrarily large vertex-stabilizers in their automorphism groups [23], the first known example of a finite half-arc-transitive (vertex- and edge-transitive but not arc-transitive) 4-valent finite graph with non-abelian vertex-stabilizer [20], and the first known examples of finite chiral polytopes of rank 5 [15]. Tn dAravshacke af the (etandard) Inv, indeyvy aenhornimne alanrithm are the fact that the finite
Table 2. The possibilities for G and T.
Table 2. The possibilities for G and T.
Table 3. Exceptional candidates. Fang et al. [13] proved that the vast majority of connected cubic Cayley graphs on non-abelian simple groups are normal.
Table 3. Exceptional candidates. Fang et al. [13] proved that the vast majority of connected cubic Cayley graphs on non-abelian simple groups are normal.
Table 1. Possibilities for X and xxx. P is a decomposition or a factorisation of I, then I’ is called a G-transitive decomposition or a G-transitive factorisation, respectively. By definition, a transitive cover, decomposition or factorisation is an isomorphic cover, decomposition or factorisation, respectively. Symmetries of decompositions have been studied in [30, 33]. In particular, Robinson conjectured that every finite group occurs as G? where (I, P) is an isomorphic factorisation of a complete graph I and G is the largest group of automorphisms of I preserving P. He showed [30, Proposition 3] that every finite group does occur as a subgroup of some G”. To our knowledge this conjecture is still open. In this paper, transitivity is required not only on the set of parts, divisors, or factors, but also on the graphs: namely on their vertices or edges or arcs. For any graph I we denote by VI, ET, AT
Table 1. Possibilities for X and xxx. P is a decomposition or a factorisation of I, then I’ is called a G-transitive decomposition or a G-transitive factorisation, respectively. By definition, a transitive cover, decomposition or factorisation is an isomorphic cover, decomposition or factorisation, respectively. Symmetries of decompositions have been studied in [30, 33]. In particular, Robinson conjectured that every finite group occurs as G? where (I, P) is an isomorphic factorisation of a complete graph I and G is the largest group of automorphisms of I preserving P. He showed [30, Proposition 3] that every finite group does occur as a subgroup of some G”. To our knowledge this conjecture is still open. In this paper, transitivity is required not only on the set of parts, divisors, or factors, but also on the graphs: namely on their vertices or edges or arcs. For any graph I we denote by VI, ET, AT
Table 1. Values of v(m) for 1 <n < 120.
Table 1. Values of v(m) for 1 <n < 120.
The following table gives the values of v(m) obtained from Theorems 11 and 12 for alln < 120. According to Nedela [28], Fujisaki used a computer search, based on the Kwak-Kwon permu- tation method (see [12] and §14), to evaluate v(n) for all m < 100. The above values of v(n) agree with all of his in this range, except in the case n = 90, where Fujisaki’s reported value v(90) = 6 appears to be a typographic error. (It is easy to construct eight non-isomorphic orientably reg- ular embeddings of Ky,» in this case.) The initial evidence provided by Fujisaki’s search was of considerable value in showing that the approach used to prove Theorem 10 was correct.
The following table gives the values of v(m) obtained from Theorems 11 and 12 for alln < 120. According to Nedela [28], Fujisaki used a computer search, based on the Kwak-Kwon permu- tation method (see [12] and §14), to evaluate v(n) for all m < 100. The above values of v(n) agree with all of his in this range, except in the case n = 90, where Fujisaki’s reported value v(90) = 6 appears to be a typographic error. (It is easy to construct eight non-isomorphic orientably reg- ular embeddings of Ky,» in this case.) The initial evidence provided by Fujisaki’s search was of considerable value in showing that the approach used to prove Theorem 10 was correct.
where A, B are represented by the vertices A, B called the vertex groups. The amalgamated sub- group H is represented by the edge H called the edge group. We can extend this to arbitrary graphs.
where A, B are represented by the vertices A, B called the vertex groups. The amalgamated sub- group H is represented by the edge H called the edge group. We can extend this to arbitrary graphs.
In general the answer is no, unless we put suitable restrictions on the amalgamated subgroups, such as: (1) the edge subgroups of a vertex group intersect trivially; (2) edge groups are restricted to cyclic subgroups; (3) in the case of polygonal products, the number of vertex groups are > 4. Applying Theorem 2.4 and using induction, we can prove:
In general the answer is no, unless we put suitable restrictions on the amalgamated subgroups, such as: (1) the edge subgroups of a vertex group intersect trivially; (2) edge groups are restricted to cyclic subgroups; (3) in the case of polygonal products, the number of vertex groups are > 4. Applying Theorem 2.4 and using induction, we can prove:
It follows that the groups of linkages of torus knots are conjugacy separable. The groups of linkages of torus knots are presented as And the group for the linkage of a torus knot with cycles within and outside the torus is pre- sented as
It follows that the groups of linkages of torus knots are conjugacy separable. The groups of linkages of torus knots are presented as And the group for the linkage of a torus knot with cycles within and outside the torus is pre- sented as
where S ranges over the representatives of similarity classes of subgroups of index 2 in A, and € does all irreducible characters of K 1 S except the principal character, and g is a fixed element ink —S. where S ranges over the representatives of similarity classes of subgroups of index 2 in A, & does all irreducible characters of K 1 S except the principal character, ys is the Mobius function, and de = CE — cE ifé € 31(S) and dg = 2cz Ce ifé € 32(S). We observe that the number Isoc®?(N;,,B;n) of equivalence classes of regular branched connected orientable n-fold coverings of a nonorientable surface N;, with branch set B is equal to
where S ranges over the representatives of similarity classes of subgroups of index 2 in A, and € does all irreducible characters of K 1 S except the principal character, and g is a fixed element ink —S. where S ranges over the representatives of similarity classes of subgroups of index 2 in A, & does all irreducible characters of K 1 S except the principal character, ys is the Mobius function, and de = CE — cE ifé € 31(S) and dg = 2cz Ce ifé € 32(S). We observe that the number Isoc®?(N;,,B;n) of equivalence classes of regular branched connected orientable n-fold coverings of a nonorientable surface N;, with branch set B is equal to
Here f2(j4) is the trace of the action of the central element C2 on the irreducible representa- tion V,, and s,(p) is, by (yet another) definition, the Schur polynomial corresponding to the partition j2, On the right, the trace can be computed by rearranging the sums: The equivalence of the two definitions of the Schur polynomials is a standard fact known as the Frobenius theorem; the proof can be found, for example, in [33]. Finally, we obtain
Here f2(j4) is the trace of the action of the central element C2 on the irreducible representa- tion V,, and s,(p) is, by (yet another) definition, the Schur polynomial corresponding to the partition j2, On the right, the trace can be computed by rearranging the sums: The equivalence of the two definitions of the Schur polynomials is a standard fact known as the Frobenius theorem; the proof can be found, for example, in [33]. Finally, we obtain
Figure 1. One graph but two different maps. The face degrees of the map on the left are 1 and 5, and for that on the right, 3 and 3. Note that a graph does not have faces at all: it has only vertices and edges.
Figure 1. One graph but two different maps. The face degrees of the map on the left are 1 and 5, and for that on the right, 3 and 3. Note that a graph does not have faces at all: it has only vertices and edges.
Figure 2. This is not a map: one of the “regions” obtained after cutting the surface along the edges of the graph is not homeomorphic to an open disk.
Figure 2. This is not a map: one of the “regions” obtained after cutting the surface along the edges of the graph is not homeomorphic to an open disk.
Figure 3. The graph K4 gives rise to a map of genus 0 (tetrahedron) having 4 faces of degree 3, and to a map of genus | having 2 faces, of degree 8 and 4 respectively.
Figure 3. The graph K4 gives rise to a map of genus 0 (tetrahedron) having 4 faces of degree 3, and to a map of genus | having 2 faces, of degree 8 and 4 respectively.
Figure 5. Maps may be considered as hypermaps whose white vertices are all of degree 2. An example of a hypermap 1s given 1n Figure 4. The above definition suggests that a hypermap is a particular case of a map. However, hypermaps were first invented as generalizations of maps [5]. Indeed, taking a map, we may insert a white vertex in the middle of every edge (see Figure 5), thus obtaining a particular case of a hypermap—such that all its white vertices are of degree 2.
Figure 5. Maps may be considered as hypermaps whose white vertices are all of degree 2. An example of a hypermap 1s given 1n Figure 4. The above definition suggests that a hypermap is a particular case of a map. However, hypermaps were first invented as generalizations of maps [5]. Indeed, taking a map, we may insert a white vertex in the middle of every edge (see Figure 5), thus obtaining a particular case of a hypermap—such that all its white vertices are of degree 2.
‘igure 6. A labeling of the hypermap of Figure 4.
‘igure 6. A labeling of the hypermap of Figure 4.
Figure 7. A map with 4 edges representing the group PSL3(2), and four maps with 6 edges representing the Mathieu group M)p. In total, there are 50 planar maps with 6 edges representing M12. A fact which rarely attracts attention is that a simple picture drawn on a piece of paper, via the triple of permutations described above, generates a permutation group. We call this group G = (o, a, ¢) the cartographic group corresponding to a (hyper)map. Of course, more often than not the group thus obtained is either S, or A,. But there exist also many other, more interesting examples. A very small sample is given in Figures 7, 8, 9.
Figure 7. A map with 4 edges representing the group PSL3(2), and four maps with 6 edges representing the Mathieu group M)p. In total, there are 50 planar maps with 6 edges representing M12. A fact which rarely attracts attention is that a simple picture drawn on a piece of paper, via the triple of permutations described above, generates a permutation group. We call this group G = (o, a, ¢) the cartographic group corresponding to a (hyper)map. Of course, more often than not the group thus obtained is either S, or A,. But there exist also many other, more interesting examples. A very small sample is given in Figures 7, 8, 9.
Figure 8. Two maps with 12 edges representing the Mathieu group M24; the one on the right was found by N. Adrianov (private communication).
Figure 8. Two maps with 12 edges representing the Mathieu group M24; the one on the right was found by N. Adrianov (private communication).
Figure 9. A hypermap with 24 half-edges representing M4 (this example is borrowed from [3]). The horizontal line is an equator on the sphere. All the 6 faces are of degree 4.
Figure 9. A hypermap with 24 half-edges representing M4 (this example is borrowed from [3]). The horizontal line is an equator on the sphere. All the 6 faces are of degree 4.
An interesting and highly nontrivial question is, given a hypermap, how to compute the corre- sponding Belyi function. Let us illustrate some initial stages of such a computation. Suppose we have the “dessin d’enfant” shown in Figure 10. [ee be Os oe ee oi Ce ob i LO SS ee E, ML A \ er Ny i i eee i Ae en Soe ae x Pew
An interesting and highly nontrivial question is, given a hypermap, how to compute the corre- sponding Belyi function. Let us illustrate some initial stages of such a computation. Suppose we have the “dessin d’enfant” shown in Figure 10. [ee be Os oe ee oi Ce ob i LO SS ee E, ML A \ er Ny i i eee i Ae en Soe ae x Pew
One more example is given in Figure 13. The corresponding Shabat polynomial is
One more example is given in Figure 13. The corresponding Shabat polynomial is
Figure 13. Applane tree obtained as a preimage ofa segment via its Shabat polynomial (courtesy of J. Bétréma). Figure 12. The star-tree and the chain-tree.
Figure 13. Applane tree obtained as a preimage ofa segment via its Shabat polynomial (courtesy of J. Bétréma). Figure 12. The star-tree and the chain-tree.
‘igure 14. Draw a tree with all its internal vertices of degree 3 and attach a loop to each leaf.
‘igure 14. Draw a tree with all its internal vertices of degree 3 and attach a loop to each leaf.
Figure 17. A decomposable map, and how it covers another map. Little squares show critical values of f. Figure 16. A cube quotiened by a rotational symmetry of order 4.
Figure 17. A decomposable map, and how it covers another map. Little squares show critical values of f. Figure 16. A cube quotiened by a rotational symmetry of order 4.
Figure 19. Iterations of the Belyi function corresponding to this “dessin d’ enfant” create a dynamical system with a complete chaos. We would like to propose here one more example to Sullivan’s theorem, see Figure 19. Our example is somewhat less explicit since we are unable to give an explicit expression of the rational function in question. Nevertheless, we may affirm that the conditions of the theorem are satisfied and that therefore this is a case of complete chaos. -m wat ec te 4 ne pisea a —_ A em le aa ene 864 ey, » ci we 4
Figure 19. Iterations of the Belyi function corresponding to this “dessin d’ enfant” create a dynamical system with a complete chaos. We would like to propose here one more example to Sullivan’s theorem, see Figure 19. Our example is somewhat less explicit since we are unable to give an explicit expression of the rational function in question. Nevertheless, we may affirm that the conditions of the theorem are satisfied and that therefore this is a case of complete chaos. -m wat ec te 4 ne pisea a —_ A em le aa ene 864 ey, » ci we 4

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  397. exp(A), the exponent of A 5. #sub(A), the number of subgroups of A 6. #r(A), the number of elements of order r 7. com(A), the order of the commutator of A
  398. #compair(A), the number of ordered pairs of commuting elements
  399. #norpair(A), the number of element-subgroup pairs (a, B) such that aBa -1 = B 10. isomdim(A), the least n such that A is isomorphic to an isometry group of euclidean space E n , alternatively the smallest n such that A has a faithful representation in GL(n) (see [2])
  400. perm(A), the smallest n such that A has a faithful permutation representation on n symbols 12. Given any group property P inherited by subgroups (e.g abelian, solvable), the 0 -1 sizing defined by P(A) = 0 if A has property P and P(A) = 1 otherwise. Most of these are sizings by definition. The parameters γ , γ , and β are sizings because of Babai's theorem [3] that given a subgroup B of the group A, any Cayley graph for A edge-contracts to a Cayley graph for B (and then use that edge-contraction cannot increase genus or Betti number). There are also many natural parameters which are not sizings:
  401. rank(A), the smallest size of a generating set for A 2. abel(A), the order of the abelianization A/A
  402. series(A), length of a composition series for A 4. transperm(A), the smallest n such that A acts faithfully and transitively on n symbols 5. pres(A), the length of the shortest presentation for A, measured by number of keystrokes in some standardized format The simple observation that every group is a subgroup of the symmetric group S n for some n explains why the first four parameters are not sizings (for the fourth, note that if A is abelian, transperm(A) = ord(a)). For pres(A), consider presentations such as A = X , Y : X 2 = Y 1000 = [Y, XYX ] = 1 with B = Y, XYX . The sum of the lengths of the relators for A are 2 + 1000 + 8. Letting U = Y , V = XYX , we get B = U , V : U 1000 = V 1000 = [U , V ] = 1 so the sum of the lengths for B is 1000 + 1000 + 4, and it is not hard to show there can be no shorter presentation for B ∼ = Z 1000 × Z 1000 .
  403. GAPS AND POLES A long standing problem for genus parameters is whether for every g there is a group of genus g. Call n a gap for the sizing s if there is no group A with s(A) = n, that is n is not in the range of s. At the other extreme, call n a pole for s if there are infinitely many groups A with s(A) = n. For the genus parameters, it is known that 0 and 1 are the only poles for γ , σ , σ o , because the Riemann-Hurwitz equation, and its generalization to embeddings of Cayley graphs [5], implies ord(A) is bounded above by 164(γ (A)-1), 84(σ o (A)-1) and 168(σ (A)-1). May and Zimmerman [7] have shown that σ o has no gaps, but it is not known whether there are any gaps for γ or σ . Conder and Tucker [4] have shown that the only possible gaps for σ occur when g ≡ 8, 14 mod(18) and there is a prime p dividing g -1 such that p ≡ 5 mod (6) and the exponent of p in the factorization of g -1 is odd. We give two more elementary examples of gaps and poles.
  404. Theorem 2.1. For the sizing #2(A), the number of involutions in A, all even positive numbers are gaps and all odd numbers, together with 0, are poles.
  405. Proof. Clearly all odd order groups have no involutions, so 0 is a pole. If A has even order, by pairing an element with its inverse, we see that the number of nonidentity, noninvolutoary elements must be even, so the number of involutions must be odd. For every odd n > 1, the dihedral group D n has n involutions and if m is odd, then #2(A × Z m ) = #2(A), so all odd numbers n > 1 are poles. The groups Z 2m for m odd all have exactly one involution, so 1 is also a pole. P REFERENCES
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