The transitive groups of degree 48 and some applications
aa r X i v : . [ m a t h . G R ] F e b THE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS
DEREK HOLT, GORDON ROYLE, AND GARETH TRACEY
Abstract.
The primary purpose of this paper is to report on the successful enumeration in
Magma of representatives of the 195 826 352 conjugacy classes of transitive subgroups of thesymmetric group S of degree 48. In addition, we have determined that 25707 of these groupsare minimal transitive and that 713 of them are elusive. The minimal transitive exampleshave been used to enumerate the vertex-transitive graphs of degree 48, of which there are1 538 868 366, all but 0 . Introduction
Since late in the 19th century, significant effort has been devoted to compiling cataloguesand databases of various types of groups, including complete lists of (representatives of theconjugacy classes of) the transitive and primitive subgroups of the symmetric groups of smalldegree. For the transitive groups, earlier references include [24, 23] (with corrections in [25]) fordegrees up to 12, [12] for degrees up to 31, [2] for the significantly more difficult case of degree32, and [11] for degrees 33 −
47. (There are 2 801 324 groups of degree 32, and a total of 501 045groups of all other degrees up to 47.) Apart from the early work of Miller, these lists have beencompiled by computer, using
GAP and
Magma .Degree 48 is once again significantly more difficult than earlier degrees, because there aremany more groups and the computations involved need more time and computer memory.The main purpose of this paper is to report on the successful enumeration of conjugacy classrepresentatives of the transitive subgroups of degree 48, which is the topic of Section 2. Thereare a total of 195 826 352 of these subgroups. The computations were carried out in
Magma andrequired a total of about one year of cpu-time. The complete list of these subgroups is availablein
Magma using an optional database that can be downloaded by users from the
Magma website. Although we carried out these computations serially on a single processor and, dueto various logistical problems, they took more than two years of real time to complete, theyare intrisically extremely parallelisable: about 98% of the cpu-time was for imprimitive groupswith blocks of size 2 and, as we shall explain shortly, that case splits into 25000 independentcalculations.We anticipate that it would be feasible to extend the catalogues up to degree 63, but thatdegree 64 will remain out of range for the foreseeable future.
Mathematics Subject Classification.
Key words and phrases. transitive group; vertex-transitive graph; census; catalogue; generator number.
MinBlockSize TransGps MinTrans Elusive2 192327620 15046 1083 3397563 10625 5904 94121 36 36 5011 0 128 1275 0 012 103 0 016 646 0 024 9 0 0Primitive 4 0 0Total 195826352 25707 713
Table 1.
Numbers of transitive, minimal transitive, and elusive groups of degree48We subsequently used our catalogue to identify those groups of degree 48 that are minimaltransitive (that is, they have no proper transitive subgroups) and those that are elusive (that is,they contain no fixed-point-free elements of prime order). We shall report on this in Section 3.The various counts of groups involved are summarised in Table 1, where the imprimitive groupshave been counted according to the smallest size of a block of imprimitivity.In Section 4, we describe the computation of the vertex-transitive graphs of order 48, alongwith some associated data.We denote the smallest size of a generating set of a group G by d ( G ). We have establishedby routine computations, that d ( G )
10 for all transitive groups of degree 48. In fact, the onlyexamples with d ( G ) = 10 have minimal block size 3 and have 10-generator transitive groups ofdegree 32 as quotients. The groups with block size 2 all satisfy d ( G ) d ( G ) have enabled the third author toremove the exceptional cases of the general bound on d ( G ) for transitive permutation group ofdegree n that he established in [28], and thereby to complete the proof of the following result,where logarithms are to the base 2. Theorem 1.1.
Let G be a transitive permutation group of degree n . Then d ( G ) (cid:22) cn √ log n (cid:23) where c := √ . (It was proved by Lucchini in [19] that this result holds for some unspecified constant c .) Sincethe proof of this result involves some lengthy case-by-case analyses, we shall just summarise itin Section 5 of this paper, and the details will be published separately by the third author [29].In a related application, the third author is now able to improve a previously unpublishedresult bounding the constant d in the result proved in [20] that d ( G ) d log n/ √ log log n forprimitive subgroups G of S n . HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 3
Notation:
For a finite group G , we will write Φ( G ), R ( G ), [ G, G ] for the Frattini subgroup,soluble radical, and derived subgroup of G , respectively. We will mostly use the notation from[31] for group names, although we simply write n for the cyclic group of order n when there isno danger of confusion.2. Computing the transitive groups of degree (see [4]) and are incorporatedinto the databases of both Magma and
GAP , and so we need only consider the imprimitivegroups. By definition, if a group G acting transitively on the set Ω of size n is not primitive,then there is at least one partition of Ω into a block system B such that G permutes the blocksof B . If we let G B denote the action of G on the blocks (“the top group”) and if B has n/k blocks of size k , then G B is a transitive permutation group of degree n/k . We say that the blocksystem B is minimal if k is minimal among block systems with k >
1. Then we can associateto each group G a set of pairs of the form { ( k, G B ) : B is a minimal block system for G with blocks of size k } . If this set contains more than one pair (imprimitive groups may of course have more thanone minimal block system), then we wish to distinguish just one of them. Thus we define the signature of an imprimitive permutation group to be the lexicographically least pair ( k, G B )associated with G , where the second component is indexed according to its order in the listof transitive groups of degree n/k already in Magma . But note that it can happen that twodifferent minimal block systems of G define the same signature.We separate the computation into parts, with each part constructing only the groups witha particular signature. Given an integer k such that 1 < k < n , and a transitive group H ofdegree n/k , the wreath product S k ≀ H contains (a conjugate of) every transitive group of degree n with signature ( k, H ). So these groups can all be found by exploring the subgroup lattice of S k ≀ H (although there are complications arising from the fact that we want representatives ofsubgroups up to conjugacy in S n .)A naive approach to the problem for a fixed k , is to deal with all candidates H simultaneously,by starting with S k ≀ S n/k , and repeatedly using the MaximalSubgroups command of
Magma ,thereby traversing the subgroup lattice downwards and in a breadth-first fashion, pruning eachbranch of the search as soon as it produces groups with signature differing from H , while usingconjugacy tests to avoid duplication. (We also have to eliminate duplicates arising from a grouppreserving more than one minimal block system with blocks of size k .) This was successfullyapplied in all cases to the transitive groups of degrees 33 −
47, and we refer the reader to [11,Section 2] for further details. In degree 48, we successfully applied this method to groups withsignatures ( k, H ) with k >
6; that is for k = 6 , , ,
16 and 24. The cpu-times in these caseswere of order 10 hours, 30 minutes, 3 minutes, 70 minutes, and a few seconds, respectively.For the examples with k = 2 , k = 2). This has recently been extended to degree 8191 by Ben Stratford, a student of the first author.
DEREK HOLT, GORDON ROYLE, AND GARETH TRACEY
The methods for k = 3 were essentially the same as in degree 36, but considerably moretime-consuming. We have G S ≀ S ∼ = C ≀ ( C ≀ S ). Let ρ be the induced projection of G onto C ≀ S . Then, since we are assuming that G is transitive, ρ ( G ) must project onto atransitive subgroup of S , and the existing catalogues contain the 1954 possibilities for thisprojection. Furthermore, either(i) ρ ( G ) is a transitive subgroup of S , in which case we can use the existing catalogue2 801 324 as a list of candidates for ρ ( G ); or(ii) ρ ( G ) is an intransitive groups of degree 32 that projects onto a transitive subgroupof S . In that case, it is not hard to show that G must be conjugate to the naturalcomplement of the base group of C ≀ H , where H is one of the 1954 transitive groupsof degree 16. (We also checked this computationally.)We enumerated the groups with k = 3 by considering each of the 2 801 324 + 1954 possibilitiesfor ρ ( G ) in turn. This involves a cohomology computation, which is analogous to that for thecase k = 2, which we shall discuss below. The computation for k = 3 took a total of about104 hours of cpu-time. Of the 3 397 563 groups on this list, ρ ( G ) is transitive for all except55 715. Similarly, for k = 4, we used the same techniques as in degrees 36 and 40, and the totalcpu-time was about 9 hours.The vast majority of the computational work was for the case k = 2, and we shall brieflyrecall how we proceed in this case. We have G W := C ≀ H , where H := G B is one ofthe groups in the known list of 25 000 transitive groups of degree 24. Again we calculate thosegroups with signature (2 , H ) for each individual group H , and the 25 000 calculations involvedare independent and could in principal be done in parallel.Let K ∼ = C be the kernel of the action of W on B . Then we can regard K as a module for H over the field F of order 2, and M := G ∩ K is an F H -submodule. We can use the Magma commands
GModule and
Submodules to find all such submodules. In fact, since we are lookingfor representatives of the conjugacy classes of transitive subgroups of W , we only want onerepresentative of the conjugation action of N := C ≀ N S ( H ) on the set all F H -submodules M of K , and we use the Magma command
IsConjugate to find such representatives.Now, for each such pair (
H, M ), the transitive groups G with H = G B and M = G ∩ K correspond to complements of K/M in H/M , and the H -conjugacy classes of such complementscorrespond to elements of the cohomology group H ( H, K/M ), which can be computed in
Magma .We also need to test these groups G for conjugacy under the action of N N ( M ). In somecases when H ( H, K/M ) is reasonably small, this can be done in straightforward fashion using
Magma ’s IsConjugate function. But in many cases this was not feasible, and we had to usethe method using an induced action on the cohomology group that is described in detail in [2,Section 2.2]. Finally, for each G that we find, we need to find all block systems with block size2 preserved by G , so that we can eliminate occurrences of groups that are conjugate in S n butarise either for distinct pairs ( H, M ) or more than once for the same pair. Again we refer thereader to [2, Section 2.2] for further details.Here are some statistical details concerning some of these calculations.
HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 5 • The numbers of groups arising from the 25000 candidates for the top group H rangesfrom 3 to 3 642 186, with average 7693 and median 778. This number is less than10000 for more than 90% of the top groups. For the majority of these groups H , thecomputations were fast. For examples, for the groups H = TransitiveGroup (24 , k ) with20000 < k H ) the total cpu-time was about 92 . G that arise is 2 963 853(about 1 .
5% of the total). • The highest dimension of a cohomology group H ( H, M ) was 26. In that case, | H ( H, M ) | =2 = 67 108 864, where elements of H ( H, M ) are represented by 26 binary digits. Thisis important because of the orbit computation on the elements of H ( H, M ). If anexample of much higher dimension than this had been encountered (which we mightexpect to be the case for a corresponding attempt to find the transitive groups of de-gree 64), then this orbit computation might not have been feasible. This occurred with H = TransitiveGroup (24 , | M | = 2 , and the pair ( H, M ) gave rise to 201 792groups G . • The case H = TransitiveGroup (24 , G ,namely 3 642 186. There were 240 possibilities for M . This case took about 34 hours ofcpu-time, using about 73GBytes RAM. • The pair (
H, M ) that resulted in the most groups G , namely 1 054 720, arose with H = TransitiveGroup (24 , | M | = 2 . Although H ( H, M ) had dimensiononly 22 in this case, there were many more orbits of the action than in the case withdimension 26 discussed above.3.
Minimal transitive and elusive groups
Minimal transitive groups.
For many applications that involve considering all possibletransitive actions of a certain degree, it is sufficient to consider only the minimal transitivegroups i.e., transitive groups with no proper transitive subgroups. (One example of this wasdiscussed in [11, Section 5], where all vertex-transitive graphs of degrees 33–47 are constructed.)Testing if a transitive group is minimal can be done by finding all of its maximal subgroups andverifying that none are transitive. As most of the groups are not minimal transitive, it provesuseful in practice to first construct some random subgroups in an attempt to find a transitiveproper subgroup, only undertaking the more expensive step of finding all maximal subgroups ifthis fails.There are a total of 25707 minimal transitive groups, all of which have minimal blocks ofsizes 2, 3 or 4 — the exact numbers of minimal transitive groups with smallest blocks of eachsize are given in Table 1.3.2.
Elusive groups.
One of the major reasons to construct catalogues of combinatorial objectsis to gather evidence relating to conjectures or other open questions. Even if a newly-constructedcatalogue does not directly contain a counterexample to a conjecture (thereby immediatelyresolving it), it can be useful in refining a researcher’s intuition regarding both the typical andextremal objects in the catalogue.
DEREK HOLT, GORDON ROYLE, AND GARETH TRACEY
A permutation group G is called elusive if it contains no fixed-point-free elements (i.e., de-rangements ) of prime order. Elusive groups are interesting because of their connection toMaruˇsiˇc’s Polycirculant Conjecture [22] which asserts that the automorphism group of a vertex-transitive digraph is never elusive . In principle, a positive resolution of the polycirculant con-jecture may simplify the construction and analysis of vertex-transitive graphs and digraphs, asit would then be possible to assume the presence of an automorphism with n/p cycles of length p for some prime p . Early catalogues of vertex-transitive graphs often used ad hoc argumentsto show that all transitive groups of the specific degrees under consideration have a suitablederangement of prime order.A permutation group G is called 2 -closed if there is no group properly containing G withthe same orbitals as G . The automorphism group of a vertex-transitive digraph is necessarily2-closed, because it is already the maximal group (by inclusion) that fixes the set of arcs ofthe digraph, which is a union of some of the orbitals. The conjecture can thus be strengthenedto the assertion that there are no elusive 2-closed transitive groups, as proposed by Klin andMaruˇsiˇc at the 15th British Combinatorial Conference [16].One might hope that there are simply no elusive groups at all, in which case both conjectureswould hold vacuously, but in fact there are a number of sporadic examples of elusive groupsand a handful of infinite families. However all the known elusive groups are not 2-closed, so donot provide counterexamples for either conjecture.It is relatively easy to test the groups for the property of being elusive by checking to seeif any of the conjugacy class representatives are derangements of prime order. For the largergroups, it is often faster to first generate some number of randomly selected elements insideeach of the Sylow subgroups in the hope of stumbling on a suitable derangement without thecost of computing all the conjugacy classes.The results of this computation reveal that there are 713 elusive groups of degree 48, withorders ranging from 5184 to 806 215 680 000. The numbers of elusive groups of degree 48 witheach minimal blocksize are given in Table 1. If an elusive group has minimal blocks of differentsizes (say 2 and 3), then it is grouped and counted according the smaller of the sizes.Of these groups 700 have a unique minimal normal subgroup, and while each of the remaining13 groups has multiple minimal normal subgroups, these minimal normal subgroups are conju-gate in S . Therefore we can partition the elusive groups according to the unique conjugacyclass of their minimal normal subgroup(s).Collectively, the 713 elusive groups share just 7 pairwise non-conjugate minimal normal sub-groups. Table 2 shows the different minimal normal subgroups that occur and the number ofelusive groups with that particular minimal normal subgroup. In addition, it gives the orderof the normalizer (in S ) of that subgroup, while the final column shows the possible minimalblock sizes that occur for that minimal normal subgroup. All but one of the possible minimalnormal subgroups are elementary abelian, but two non-conjugate (but obviously isomorphic)groups of orders 2 and 3 occur. For example, the first row shows that an elusive group withminimal normal subgroup C either has minimal blocks of size 2 (only) or minimal blocks ofsizes both 2 and 3. HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 7
Group | Normalizer | Frequency Min. Blocks C · { } or { , } C · { } C · { } C · { } C · · { } C · · · { } A(6) · · { } Table 2.
Minimal normal subgroups of elusive groups of degree 48With this many elusive groups, and no obvious way to get a compact description, it wouldseem unlikely that the Polycirculant Conjecture can be proved by first classifying elusive groups.4.
Vertex-transitive graphs of order 48
The class of vertex-transitive graphs plays a central role in algebraic graph theory, oftenproviding extremal cases or illuminating examples in the study of many graphical properties.Although it is not strictly necessary to have a complete list of the transitive groups of degree d in order to compute a complete list of vertex-transitive graphs of order d , it is conceptuallysimple to compute all the vertex-transitive graphs from the transitive groups.For notational convenience, we say that a graph Γ is G -vertex-transitive (or just G -transitive)if G Aut(Γ) and G acts transitively on V (Γ). Given a list of all the transitive groups of somefixed degree, in principle it suffices to consider each group G in turn, construct all the G -transitive groups, and then merge the lists from the different groups, removing all but oneisomorphic copy of each graph.As stated, this naive algorithm would do far too much work, constructing large numbersof isomorphic copies of most of the graphs. However we can reduce this work in two ways.First we can restrict our attention to the minimal transitive groups, because if H G and H is transitive, then any G -transitive group is H -transitive. Secondly, we can do some workto avoid constructing graphs that are obviously isomorphic to ones that have been, or will be,constructed elsewhere.The transitive groups of degree 48 and order 48 are necessarily minimal transitive, and wedeal with these separately from the larger minimal transitive groups. This separates out thevertex-transitive graphs with a regular subgroup of automorphisms, i.e., the Cayley graphs, fromthe remainder. This is common practice, because there are a number of interesting questionsand conjectures where the distinction between Cayley graphs and non-Cayley graphs appearsto be subtle but significant. DEREK HOLT, GORDON ROYLE, AND GARETH TRACEY
Cayley graphs.
Given a group G , and a set of group elements C ⊆ G such that G / ∈ C and C − = C , the Cayley graph
Cay(
G, C ) is the graph defined as follows: V (Cay( G, C )) =
G,E (Cay(
G, C )) = { ( g, cg ) | g ∈ G, c ∈ C } . It is immediate that G is a regular subgroup (acting by right-multiplication) of the auto-morphism group of Cay( G, C ), and it is well-known that any vertex-transitive graph whoseautomorphism group has a regular subgroup R is a Cayley graph for R . The set C is calledthe connection set for the Cayley graph and it is precisely the neighbourhood of the vertex G . (The conditions on C are simply to ensure that the resulting graphs are undirected andloopless.)While all Cayley graphs are vertex transitive, not all vertex-transitive graphs are Cayleygraphs, with the canonical example here being the Petersen graph. For small orders, the vastmajority of vertex-transitive graphs are Cayley graphs, but it is not known if this holds ingeneral. In other words, is it true that the proportion of vertex-transitive graphs of order atmost n that are Cayley graphs tends to 1 as n increases?If we define Ω = {{ g, g − } | g ∈ G, g = G } then every subset of Ω determines the connectionset for some Cayley graph of G and vice versa. If G has a involutions, and b non-identityelement-inverse pairs, then | Ω | = a + b , and so there are exactly 2 a + b Cayley graphs for G . Aswe are almost always only interested in isomorphism classes of graphs, we need to remove, orpreferably never construct, all but one representative of each isomorphism class. There is oneobvious source of isomorphisms, namely those arising from the automorphism group of G . Moreprecisely, if σ ∈ Aut( G ) then Cay( G, C ) is isomorphic to Cay(
G, C σ ). So if we define a Cayleyset to be an orbit of Aut( G ) acting on the set of subsets of Ω, then it suffices to consider justone connection set from each Cayley set.In practice, we fix an arbitrary order on Ω, use GAP to compute the action of Aut( G ) on Ω,and then a simple orderly-style algorithm to compute the lexicographically least representativeof each Cayley set. This makes heavy use of Steve Linton’s SmallestImageSet [18] to ensurethat only lexicographically-least subsets of Ω are considered at every stage. The theoreticalnumber of Cayley sets for G can be determined by calculating the cycle index polynomial ofAut( G ) acting on Ω (using the undocumented CycleIndexPolynomial function in
Magma )and applying P´olya’s Enumeration Theorem. For each of the 52 groups examined, the actualnumber of Cayley sets constructed by the orderly algorithm matches the theoretical number,giving us a high degree of confidence in this stage of the computation.Using the list of (representatives of) Cayley sets we next construct the corresponding listof Cayley graphs. Although this list is free of isomorphisms induced by the action of Aut( G )on the connection sets, there can be additional isomorphisms. Therefore we filter the list ofCayley graphs for each group, removing any graph that is isomorphic to an earlier graph in thelist. Sometimes there are no isomorphisms except the ones induced by Aut( G ), and so the finalfiltering step does not remove any graphs. Groups with this property are called CI -groups andthere is a substantial literature on the still-open question of characterizing CI -groups. HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 9
There are 52 groups of order 48 and the results of the Cayley graph computations for thosegroups are given in Table 3. The groups are numbered from 1 to 52 according to their order inthe small group libraries of
Magma and
GAP (the groups are in the same order in each library).The group structure is the description returned by the
GAP command
StructureDescription ,the values a and b are the number of involutions and the number of non-identity element-inversepairs respectively. The column labelled | Aut | lists the order of the automorphism group of thegroup. In most cases, the value 2 a + b / | Aut | is an approximation (an underestimate) for thenumber of Cayley sets for that group. Where an entry in the | Aut | column is marked withan asterisk (such as ∗
192 for group number 8), this indicates a group where Aut( G ) does notact faithfully on Ω. These groups are characterized by the existence of a group automorphism σ ∈ Aut( G ) such that g σ ∈ { g, g − } for each element g . It is known that such a group is eitheran abelian group (where the inverse map is a group automorphism) or a generalised dicyclicgroup (Watkins [30]). In each of the cases indicated in Table 3, the kernel of the action ofAut( G ) on Ω has order 2, and so these groups yield approximately 2 a + b +1 / | Aut | Cayley sets.The column “Cayley Sets” gives the exact number of Cayley sets obtained from Polya’sEnumeration Theorem, while the final column “Cayley Graphs” gives the actual number ofpairwise non-isomorphic Cayley graphs. This last step is computationally non-trivial becauseof the sheer size of some of the lists. For example, there are more than 360 million Cayleygraphs for the most prolific group ( C × D ).A group is a CI-group if and only if the number of Cayley graphs is equal to the number ofCayley sets, so the table shows that the abelian group C × C × C × C is the only CI-groupof order 48. Table 3.
Cayley graphs on 48 vertices
No. Structure a b | Aut | Cayley Sets Cayley Graphs1 C : C C ∗
16 2151936 21229443 ( C × C ) : C C × S C : C C : C
13 17 96 13641984 118802407 D
25 11 192 364086016 3607161128 C : Q ∗
192 275712 2556969 C × ( C : C ) 3 22 96 647168 59764810 ( C : C ) : C C × ( C : C ) 3 22 192 454176 37070412 ( C : C ) : C C : C ∗
192 586752 48494414 ( C × C ) : C
15 16 96 28924416 2313984815 ( C × D ) : C
17 15 96 47661696 4585552016 ( C : Q ) : C C × Q ) : C
13 17 96 12473472 1195227218 C : Q C × C ) : C C × C ∗
192 452032 43180821 C × (( C × C ) : C ) 7 20 64 3373440 287619222 C × ( C : C ) 3 22 64 1081344 904064 C × C ∗
32 2336768 218323224 C × ( C : C ) 3 22 32 1603584 149900825 C × D C × QD C × Q C .S = GL(2 , .C ,
3) 13 17 48 22758528 2258939230 A : C C × A C × GL(2 ,
3) 3 22 48 808832 79133833 (( C × C ) : C ) : C C × ( C : Q ) 3 22 ∗
384 386784 27961635 C × C × S
15 16 192 16697472 1415952836 C × D
27 10 384 373069248 36245853637 ( C × C ) : C
15 16 96 29701632 2239560838 D × S
23 12 96 383280384 34981500839 ( C × S ) : C
11 18 96 9219840 545276040 Q × S C × S ) : C
19 14 288 34347520 3075948042 C × C × ( C : C ) 7 20 ∗ C × (( C × C ) : C ) 19 14 192 54719616 4764117644 C × C × C ∗
384 967808 80665645 C × D
11 18 128 6653184 565476846 C × Q C × (( C × C ) : C ) 7 20 96 2397184 186521648 C × S
19 14 48 182656512 17792370449 C × C × A
15 16 144 15715936 1530670050 ( C × C × C × C ) : C
15 16 5760 413344 41124851 C × C × C × S
31 8 8064 72984704 7103969652 C × C × C × C
15 16 ∗ The total number of Cayley graphs, after removing isomorphs both within and between the52 lists of graphs is 1 536 366 616, approximately 1 .
54 billion Cayley graphs of order 48.4.2.
Non-Cayley graphs.
The automorphism group of a vertex-transitive graph that is not aCayley graph is a transitive group that has no regular subgroups. Thus the simplest approachis to consider each minimal transitive group G in turn, compute all the G -transitive graphsand then remove both unwanted isomorphs and any Cayley graphs that have accidentally beenconstructed along the way.If G is a transitive group then its orbitals are defined to be the orbits of G on V ( G ) × V ( G ).If O = ( x, y ) G is an orbital of G , then ( y, x ) G is also an orbital of G , called the paired orbital of O . A graph is G -transitive if and only if its edge-set is the union of pairs of orbitals of G (identifying an edge xy with a pair of oppositely-directed arcs { ( x, y ) , ( y, x ) } ).An orbital graph of G is a graph whose edge-set is the union of a single pair of orbitals.Let G ′ be the intersection of the automorphism groups of all the orbital graphs of G . (Thisis essentially an undirected version of the 2-closure of a group, sometimes called the strong -closure of the group.) Then any G -transitive graph is G ′ -transitive and so it is only necessaryto process G ′ . In itself, this does not reduce the amount of work required because, by definition, G and G ′ have precisely the same pair-closed sets of orbitals to consider. However it is often HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 11 the case that G ′ and H ′ are conjugate even when G and H are not. Therefore by constructingthe strong 2-closure of all of the minimal transitive groups and then throwing out all-but-onefrom each conjugacy class, we end up with a much smaller list of groups to process. This listcan be even further reduced by noting that G ′ sometimes has a regular subgroup although G ,again by definition, does not. In this situation, every G ′ -transitive graph is a Cayley graph, andas these have already been constructed, there is no need to process G ′ .The minimal transitive groups of order greater than 48 collectively have 840 pairwise non-conjugate strong 2-closures. It is easy to verify that a small group has no regular subgroupsbefore processing it further, but for the larger groups this becomes too time-consuming. Howeverthe larger groups tend to have few orbitals, and so it is easy to construct all of the transitivegraphs for these groups. The resulting list of graphs then contains all of the non-Cayley graphs,but also many Cayley graphs that must be removed. Due to the sheer size of the computation,this is a rather lengthy and somewhat intricate process, but on completion we end up with2 501 750 non-Cayley graphs, of which 2 501 630 are connected. Hence the total number ofvertex-transitive graphs on 48 vertices is 1 538 868 366, of which just 0 . Edge-transitive and half-arc transitive graphs.
A graph is called edge-transitive ifits automorphism group is transitive on edges (i.e., unordered pairs of adjacent vertices) and arc-transitive if it is transitive on its arcs (i.e., ordered pairs of adjacent vertices). An edge-transitive graph might also be vertex-transitive, but there are edge-transitive graphs that arenot vertex transitive, in fact some that are not even regular. Conder & Verret [3] have computedall of the edge-transitive graphs on up to 47 vertices, separately finding those that are vertextransitive, and those that are not.As a result of the computations reported above, we can go one step further and find theedge-transitive graphs of order 48 that are also vertex-transitive. Thus from the 1 .
54 billionvertex-transitive graphs of order 48, we extracted 189 edge-transitive graphs, of which 115 areconnected, 115 (sic) are twin-free ( twins are vertices with the same neighbourhood) and just 71are both connected and twin-free.We can also extract a few more interesting graphs from our lists. A graph is called half-arc transitive (or just half-transitive ) if it is vertex transitive and edge transitive, but not arctransitive. The most famous, and smallest, such graph is the 4-regular graph on 27 verticesknown as the
Doyle-Holt graph after its independent discoverers Doyle [6] (originally in anunpublished Masters Thesis at Harvard in 1976) and Holt [10] in 1981.The data tabulated in Conder & Verret indicate that there is a single half-arc-transitive graphon 27 vertices (degree 4), 2 on 36 vertices (of degrees 8 and 12), 2 on 39 vertices (degrees 4 and8) and 3 on 40 vertices (all of degree 8). To this we can add another 4 half-arc-transitive graphson 48 vertices (all of degree 8).All four of these 8-regular half-arc-transitive graphs are Cayley graphs for at least one groupof order 48, with the groups occurring being ( C × C ) : C (Group 3 from the list above), A : C (group 30), C × C × A (group 49) and ( C × C × C × C ) : C (group 50). Maximal generating number of transitive groups of degree 48
For an arbitrary group G , let d ( G ) be the minimal size of a generating set of G . As wesaw earlier, for most of the transitive groups G of degree 48 that are imprimitive with blocksize 3, the quotient group ¯ G := G/O ( G ) of G is naturally isomorphic to a transitive groupof degree 32, There are five such groups with d ( ¯ G ) = 10, namely TransitiveGroup (32 , i ) for i ∈ { , , , , } , and it turned out that there are also fivecorresponding groups G , also with d ( G ) = 10. A lengthy but routine computation showed thatthese are the only transitive groups of degree 48 with d ( G ) > G with minimal block size 2, there are 11 groups with d ( G ) = 9, and thesehave signatures (2 , H ), where H = TransitiveGroup (24 , i ) with i ∈ { , , } . Themaximum value of d ( G ) among primitive groups and groups with minimal block size at least 4is 6, which arises with block sizes 4 and 6 only.In [28], the problem of finding numerical upper bounds for d ( G ) for an arbitrary transitivepermutation group G of degree n is considered. It had already been proved in [19] that d ( G ) isat most cn √ log n in this case, where c is an unspecified absolute constant. This bound is shownto be asymptotically best possible in [17] (that is, there exists constants c , c , and an infinitefamily ( G n i ) ∞ i =1 of transitive groups of degree n i , with c n i d ( G ni ) √ log n i c for all i ).In [28] it is proved that, apart from a finite list of possible exceptions, the bound d ( G ) j cn √ log n k holds, where c := √ (and logarithms are to the base 2). This bound is best possiblein the sense that d ( G ) = √ n √ log n = 4 when G = D ◦ D < S and n = 8, although it seemslikely that there are better bounds that hold for sufficiently large n .The information in the first paragraph above concerning generator numbers in transitivegroups of degree 48 has helped the third author to complete the proof of Theorem 1.1 therebydispensing with the finite list of exceptions. There are, however, a number of other steps in thisproof, some of which involve lengthy case-by-case analyses. For this reason, we will just outlinethe general strategy of the proof in this paper, and the details will be published separately bythe third author.First, by [28, Theorem 5.3], one only needs to prove Theorem 1.1 when G is imprimitive withminimal block size 2, and n has the form n = 2 x y y = 0 and 17 x
26; or y = 1and 15 x
35. Thus, in particular, G may be viewed as a subgroup in a wreath product2 ≀ G Σ , where Σ is a set of blocks for G of size 2. It follows that d ( G ) d G Σ ( M ) + d ( G Σ ),where M is the intersection of G with the base group of the wreath product, and d G Σ ( M ) isthe minimal number of elements required to generate M as a G Σ -module. With this reductionin mind, the proof of Theorem 1.1 is comprised of two main ingredients: upper bounds on d G Σ ( M ), and upper bounds on d ( G Σ ). We summarise the third author’s approach to these twosub-problems in the next few paragraphs.We note first that the bulk of the proof is taken up with finding upper bounds on d G Σ ( M ).Since d G Σ ( M ) d H ( M ) for any subgroup H of G Σ , the strategy of the third author in [28] inthis case involved replacing G Σ by a convenient subgroup H of G Σ , and then deriving upperbounds on d H ( M ), usually in terms of the lengths of the H -orbits in Σ. This approach turns outto be particularly fruitful when H is chosen to be a soluble transitive subgroup of G Σ , whenever HE TRANSITIVE GROUPS OF DEGREE 48 AND SOME APPLICATIONS 13 such a subgroup exists. When G Σ does not contain a soluble transitive subgroup, however, theanalysis becomes much more complicated. This led to less sharp bounds, and ultimately, theomitted cases in [28, Theorem 1.1].The new approach to bounding d G Σ ( M ) involves a careful analysis of the orbit lengths ofsoluble subgroups in a minimal transitive insoluble subgroup of G Σ , building on the work in[28] in the case n = 2 x
3. This analysis, together with upper bounds on d H ( M ) (for soluble H G Σ ) in terms of the lengths of the H -orbits in Σ, is then used to derive an upper boundfor d H ( M ). An upper bound for d G Σ ( M ) follows.The second sub-problem is to find an upper bound for d ( G Σ ). The group G Σ is a transitivepermutation group of degree n/ x − y
5, where n , x , and y are as above. The upperbound d ( G Σ ) c n √ log n can be derived by using induction on n . However, combining this withthe upper bounds on d ( G Σ ) detailed in the previous two paragraphs is not enough to proveTheorem 1.1 in all of the required cases. Therefore, a more careful approach is required. Thisapproach was used in the proof of Lemma 5.12 in the third author’s paper [28]. Informally, theidea is as follows. There exists a factorisation n = r . . . r t of n such that either(1) d ( G Σ ) P i = 1 t − d ( r i , r i +1 . . . r t ) + log r t ; or(2) Either r t
32, or r t = 48, and d ( G Σ ) P i = 1 t − d ( r i , r i +1 . . . r t ) + d trans ( r t ).Here, d trans ( m ) := max { d ( X ) : X transitive of degree m } . If 2 m
32, or if m = 48, thenwe know d trans ( m ) precisely, by [2], and this paper, respectively.The function d ( r, s ) is defined as the maximum of d X ( K X (∆)), where(i) X runs over the transitive permutation groups of degree rs with minimal block size r ;(ii) ∆ runs over the blocks for X of size r ;(iii) K X (∆) is the kernel of the action of X on ∆; and(iv) d X ( K X (∆)) is the minimal number of elements required to generated K X (∆) as a normalsubgroup of X .Upper bounds on d ( r, s ) are available from [28]. Thus, we can find upper bounds on d ( G Σ )by going through all factorisations of n , and taking the maximum of the bounds coming from(1) and (2) above. These maximums almost always come from (2). Thus, the new result d trans (48) = 10 from this paper plays a vital role in deriving upper bounds on d ( G Σ ), whencesolving the second sub-problem in the proof of Theorem 1.1. Acknowledgements.
We would like to thank Michael Giudici for a number of helpful discus-sions on transitive permutation groups.
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Mathematics Institute, University of Warwick
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