A Character Condition for Quadruple Transitivity

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 757134,13pages

doi:10.1155/2011/757134

Research Article

A Character Condition for Quadruple Transitivity

S. Aldhafeeri and R. T. Curtis

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Correspondence should be addressed to S. Aldhafeeri,[email protected]

Received 9 May 2011; Accepted 22 June 2011

Academic Editor: Frank Werner

Copyrightq2011 S. Aldhafeeri and R. T. Curtis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let G be a permutation group of degree n viewed as a subgroup of the symmetric group S∼Sn.

We show that if the irreducible character of S corresponding to the partition of n into subsets of sizesn−2 and 2, that is, to say the character often denoted byχn−2,2_{, remains irreducible when}

restricted to G, then n4, 5 or 9 and G∼S3, A5, or PΣL28, respectively, or G is 4-transitive.

1. Introduction

LetGbe a permutation group of degreen. The connection between the multiple transitivity of

Gand the irreducibility of certain irreducible characters of of the symmetric group Snwhen

restricted toG goes back to Frobenius, and a proof of his classical result can be found in Tsuzuku1. In2, Saxl strengthens this result, limiting the irreducibility hypothesis to the charactersχn−k,k_G , a special case of his theorem having been proved by Neumann3. In4 Saxl uses the Classification of Finite Simple Groups to go much further. He obtains a complete list of pairsχ, Gwhereχis an irreducible character of the symmetric group Sn, andGis a

subgroup of Snsuch that the restriction ofχtoGremains irreducible.

The present paper is concerned with the case whenχn−2,2_G is irreducible, proving that

G is 4-transitive except in three small cases which are worked out explicitly. As such, the result, which appears as Theorem3.3, can be deduced from2,3,5. However, the approach used here is self-contained and elementary.

2. Notation and Preliminary Results

2.1. Permutation Characters

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a basis of ann-dimensional complex vector space, which becomes aCG-module when we defineeiνeνi, forν∈G, and extend linearly. IfπXdenotes the character aﬀorded by this

representation and the permutationνhas cycle shape 1r_.₂s_{. . .}_{when written as a product of}

disjoint cycles, then

πXν r |FixXν|, 2.1

where FixXνdenotes the set of points ofXfixed by the permutationν. The subscriptXwill be omitted when there is no fear of confusion. The standard inner product defined on the space of class functions when restricted togeneralizedcharacters becomes

χ, ψ 1

|G|

g∈G

χgψg, _2.2

whereχ andψ are characters of the groupG, andαdenotes the complex conjugate of the complex numberα. The trivial character ofGwhich takes the value 1 on every element ofG

will be denoted by 1G. Then we have the following result; it is often referred to as Burnside’s Lemma but is actually due to Cauchy and Frobenius.

Theorem 2.1. IfGis a permutation group acting on the setXwith permutation characterπX, then we have the following result:

πX,1GorbG, X, 2.3

where orbG, Xdenotes the number of orbits whenGacts onX.

Proof. Firstly note that

a, ν|

a∈X ν∈G , a

ν_a

a∈X

|StabGa|

ν∈G

|Fixν|, 2.4

where StabGadenotes the stabilizer inGofa. So, ifGhas orbitsO1, O2, . . . , Omwhen it acts

onX, the Orbit-Stabilizer theorem implies that

a∈X

|StabGa|

m

i1

ai∈Oi

|StabGai|

m

i1

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wherebiis a representative of the orbitOi. Thus,

πX,1G

1

|G|

ν∈G

|Fixν| ·1 1

|G||G|mm. 2.6

If the groupGacts on the setXand on the setY with permutations charactersπXand

πY, respectively, thenGalso acts on the Cartesian productX×Y by defining

x, yνxν, yν. 2.7

An ordered pair is only fixed byνif both, its components are fixed, and so we see that

|FixX×Yν||FixXν| · |FixYν|. Then we have,

Theorem 2.2. If the groupG acts on the sets X and Y with permutation charactersπX and πY respectively, then

πX, πYorbG, X×Y. 2.8

Proof.

πX, πY

1

|G|

ν∈G

|FixXν| · |FixYν|

1

|G|

ν∈G

|FixXν|.|FixYν|

_|1

G|

ν∈G

|FixX×Yν|orbG, X×Y.

2.9

Corollary 2.3. IfGacts transitively onX anda∈X, thenπX, πY orbGa, Y, the number of

orbits of the stabilizer of a point inXonY.

Corollary 2.4. IfGacts transitively onXanda∈X, thenπX, πXorbGa, X, the rank of the

permutation action ofGonX.

In the case when S ∼ Sn, the symmetric group acting on an n-element set Ω {1,2, . . . , n}, we denote the associated permutation character byπ1. More generally πk will

denote the permutation action ofS ∼ Sn on unorderedk-element subsets. In particular,π2

will denote the permutation character of the action ofSonΔ {{a, b} |a, b ∈Ω}, the set of unordered pairs of elements ofΩwhich we shall call duads. Thus, ifν ∈Shas cycle shape 1r_.₂s_{. . .}_{, then}

π1ν r, π2ν

r

2

s. 2.10

By Theorem2.1and its corollaries we see that

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since the stabilizer of a point clearly has two orbits on duads, and the stabilizer of a duad has three orbits on duads, namely, the orbit containing just the fixed duad,δsay, the orbit containing duads which intersectδin one point, and the orbit of duads disjoint fromδ. We thus have that

π11Gχ, π21Gχψ, 2.12

where χ and ψ are distinct irreducible characters of S of degrees n −1 and nn −3/2, respectively.

2.2. The Irreducible Characters of the Symmetric Group S

n

Definition 2.5. A partitionλof the integernis a sequenceλ λ1, λ2, . . . , λkwhereλi ≥λi1

fori1, . . . , k−1 andk_i1λin.

The irreducible characters of Snare in one-to-one correspondence with the partitions of

nand we denote the character corresponding toλbyχλ_{. In this notation, the trivial character}

of Sncorresponds to the partition ofninto a single subset of sizenand is thus denoted by

1S χn. The permutation character of degreenis given byπ1 χnχn−1,1. Indeed we

have the general result.

Theorem 2.6. Ifk≤n/2 andπkdenotes the permutation character of the action of Snon unordered

k-subsets, then

πkχnχn−1,1· · ·χn−k,k. 2.13

That this permutation character is a sum of irreducibles of the right degree is readily shown and a proof may be seen in James and Liebeck6, page 344. That these irreducibles are indeed those corresponding to the above partitions may be verified using the celebrated Murnaghan-Nakayama rule, see7, page 79, which enables one to evaluate the character value of the irreducibleχλon any conjugacy class of the symmetric group. We see that for

k≤n/2 we have

χn−k,kπk−πk−1, 2.14

and so, ifσ∈Sn, then

χn−k,kσ k-subsets fixed byσ−k−1-subsets fixed by σ. 2.15

In particular, the degree ofχn−k,kis given by

χn−k,k1

n

k

−

n

k−1

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Applying2.15to a permutationσwith cycle shape 1r_.₂s_{. . .}_{, we have}

χn−2,2σ

r

2

s−r rr−3

2 s. 2.17

2.3. Multiple Transitivity

Definition 2.7. The groupG is said to act k-transitively on the set X if, and only if, given

x1, x2, . . . , xkandy1, y2, . . . , xk, two orderedk-tuples of distinct points ofX, there exists a

permutationσ∈Gsuch thatxσ_i yifori1· · ·k.

Definition 2.8. The groupGis said to actk-homogeneously on the setXif, and only if, given

A, B⊂Xwith|A||B|k, there exists a permutationσ∈Gsuch thatAσ_B_.

In other words, G is k-homogeneous if it acts transitively on the k-subsets. Those fascinating groups which are k-homogeneous without beingk-transitive have been investigated by Kantor, see 8,9, and the Livingstone-Wagner theorem of 10 is a highly significant result in this area. In this paper, we shall be concerned with the cases k

2,3, and 4 and thus with the action of subgroups of Snon unordered pairs of points, which

we shall refer to as duads, and on triples and tetrads. We can immediately see that the following applies.

Lemma 2.9. A group which is 2-homogeneous without being 2-transitive must have odd order.

Proof. Suppose thatGhas even order and acts 2-homogeneously onX. Letα{a1, a2}and

β{b1, b2}be two duads; we must produce a permutation which mapsaitobifori1 and 2.

By 2-homogeneity, there exists aσ∈Gsuch thatασ _β_{, and since}_G_{is even and again using}

the 2-homogeneity, there exists an elementτ ∈Gof order 2 which acts asτ b1 b2· · ·. If

aσ₁ b1, then we are done, otherwiseστwill suﬃce.

2.4. Restriction to a Subgroup of S

n

In what follows, the subgroupGwill be a fixed subgroup ofS∼Sn, and we shall denote the

restriction of the characterχofStoGby

χ|G. 2.18

In order to simplify the notation, we shall denote

π1|G by ψ1, π2|G by ψ2, χn−1,1|G by ψn−1,1, andχn−2,2|G byψn−2,2. 2.19

So we shall have

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The restriction to G of an irreducible character of S may or may not remain irreducible. However, ifG≤S∼Sn, then using Theorems2.1and2.2we have

1ψ1,1Gis the number of orbits ofGon points,

2ψ2,1Gis the number of orbits ofGon duads,

3ψ1, ψ1is the number of orbits ofGonΩ×Ω,

4ψ1, ψ2is the number of orbits ofGonΩ×Δ, and

5ψ2, ψ2is the number of orbits ofGonΔ×Δ.

Note that ifχandψ are as in2.12, thenχ|G ψn−1,1andψ|G ψn−2,2. We shall use their correct labels in what follows but shall not make use of the correspondence with partitions in our argument.

2.5. General Preliminary Results

A result which will be appealed to several times in what follows is the following.

Lemma 2.10. The only permutation which commutes with a transitive permutation group and which

fixes a point is the identity. In particular, a transitive, abelian permutation group acts regularly.

Proof. LetKbe a transitive subgroup of the symmetric group Snacting on the setΩ, and let

the permutationσ ∈ Sn commute with K and fix a point a ∈ Ω. ThenF Fixσ ⊂ Ωis

invariant under the action ofKand is thus the empty setφorΩ. Buta∈F /φ, and soF Ω

andσ1.

IfKis abelian andσ∈Ka, the stabilizer inKofa, thenσcommutes withKand fixes

the pointa. Thus,σ 1, from the above, and soKa 1, and the transitive groupKacts

regularly.

A well-known result about linear groups has been taken from Theorem 7.3 of Huppert

11, page 187.

Theorem 2.11. The general linear group GLmpf_{contains a cyclic subgroup}_K_{of order}_pmf₋_{1. We}

haveCGK K and thatNGK/Kis cyclic of orderm. (Such cycles of maximal possible length

are referred to as Singer cycles.)

3. Statement and Proof of Main Theorem

LetG≤Sn. The transitivity ofGis related to the reducibility of certain irreducible characters

ofSupon restriction toGas follows.

Lemma 3.1. G is doubly transitive⇔ψn−1,1 ∈IrrG.

Proof. Follows immediately from Corollary2.4.

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Proof. If G is 4-transitive, then it is certainly doubly transitive, and so ψn−1,1 ∈ IrrG. Moreover, ifδis a duad, thenGδcertainly has just three orbits on duads, as described towards

the end of Section2.1. Thus,

3ψ2, ψ2

1Gψn−1,1ψn−2,2,1Gψn−1,1ψn−2,2

≥2ψn−2,2, ψn−2,2, 3.1

and soψn−2,2, ψn−2,21 andψn−2,2∈IrrG.

It is thus natural to ask in what circumstances we can haveψn−2,2 ∈IrrGwithout

G being 4-transitive. Our main result is the following theorem which appeals to the Odd-Order theorem of Feit and Thompson and uses part of an argument due to Kantor8but is otherwise self-contained and based on Aldhafeeri12.

Theorem 3.3. LetGbe a subgroup of the symmetric group of degreenwithn≥4, thusG≤S∼Sn,

and letχn−2,2represent the irreducible character ofScorresponding to the partition ofninto a pair

and a subset of cardinalityn−2. Then

χn−2,2|G∈IrrG 3.2

(i.e., the restriction ofχn−2,2toGis an irreducible representation ofG) if, and only if,

in4 andG∼S3, iin5 andG∼A5,

iiin9 andG∼PΣL28, or

ivGis 4-transitive onnletters.

Proof. Note that the characterχn−2,2of Snonly exists forn≥4, and so the restriction onnis

only required to make the theorem meaningful. In casei, the subgroupGis not transitive, but no restriction was placed on the transitivity ofG.

We shall assume that G is not 4-transitive and that ψn−2,2 ∈ IrrG. We have

ψn−2,21 nn−3/2≥2 forn≥4. Thus,

ψ1,1G

ψ2,1G

1ψn−1,1,1G

, 3.3

and so orbG,Ω orbG,Δ k, say. But if orbG,Ω kthen orbG,Δ≥k 2

, as we may choose a duad from each of two orbits ink

2

ways. Nowk 2

> k fork > 3, and ifk 3, at least one of the three orbits must have more than one point in itsince n ≥ 4, and so orbG,Δ≥ 3

2

1 4. Thus,k≤ 2. Ifk 2 and both orbits are of cardinality greater than 1 then orbG,Δ≥3, a contradiction, so we must have one orbit of length 1 and the other of lengthn−1.

Nowψn−2,21 nn−3/2> n−1ψn−1,11providedn2₋₅_n₂_>_{0, which holds}

forn≥5. So providedn >4, we haveψn−2,2, ψn−1,10. Thus,

orbG,Ω×Ω ψ1, ψ1

ψ1, ψ2

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Table 1

A4 12 4 3 3

14 ₂2 _1.3 _1.3

ψ2,2 ₂ ₂ ₋₁ ₋₁

1 1 ω ω

Table 2

D8 8 8 4 4 4

14 ₂2 ₄ ₂2 ₁2_.₂

ψ2,2 ₂ ₂ ₀ ₀ ₂

1 1 1 1 1

1 1 −1 −1 1

Table 3

S3 6 3 2

14 _1.3 ₁2_.₂

ψ2,2 ₂ ₋₁ ₀

ψ1 4 1 2

For this reason, we treat n 4 separately and consider the circumstances in which this 2-dimensional representation of S4 can remain irreducible when restricted to a subgroup G.

Since all irreducible representations of abelian groups are linear, we must have that G is nonabelian, and the only proper non-abelian subgroups of S4are A4,D8, and S3, fixing a point.

In the following diagram we show the restrictions ofχ2,2 to subgroups of S4 isomorphic to

A4,D8, and S3, respectively. The top line in Tables1,2, and3gives the order of the centralizer

of elements in the conjugacy class of that column, and the second row gives the cycle shape of elements in that classviewed as elements in S4. It is clear that in the A4 and D8 cases

the restricted character is the sum of the two linear characters shown, but in the S3 case the

restricted character is indeed irreducible.

We may now assume thatn≥5 and that3.4holds. Let us return to the case reached above whereGhas two orbits onΩ, of lengths 1 andn−1, respectively. In order to dismiss this caseforn >4, we may assume thatG∼Sn−1, as a character which remains irreducible when

restricted to a subgroup and certainly remains irreducible when restricted to a supergroup of that subgroup. Suppose the two orbits ofGare{a}and{b1, b2, . . . , bn−1}. But now orbG,Ω × Ω 5 with orbit representatives{a, a,a, b1,b1, a,b1, b1,b1, b2}. However, orbG,Ω × Δ 6 with orbit representatives

{a,{a, b1},a,{b1, b2},b1,{a, b1},b1,{a, b2},b1,{b1, b2},b1,{b2, b3}}. 3.5

Note that whenn 4 and G ∼ S3, we do indeed have ψ1, ψ1 5 and that ψ1, ψ2 ψ1, ψ1ψ2,26. However, ifn >4, this contradicts3.4, and so we must have thatGacts

transitively onΩand onΔ.

Now let Ω {a, b, c, . . .}. The transitivity of G together with 3.4 imply that the stabilizer of a point, Ga, has the same number of orbits on points as it has on duads. As

above, ifGahaskorbits onΩ, then it has at least

_k

2

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2≤k≤3. Ifk3, then at least one of the orbits ofGamust have length greater than 1since

n≥5, and so the number of orbits ofGaon duads is at least 4, a contradiction. Thus,k 2

andGis doubly transitive. In particular,ψn−1,1, ψn−2,2∈IrrG, and we have

orbG,Ω×Δ 2, orbG,Δ×Δ 3. 3.6

SinceGδ, the stabilizer inGof the duadδ {a, b}, has just two orbits onΩ, these must be {a, b}and{c, d, . . .}, and soGδacts transitively onΩ\ {a, b}. The transitivity ofGon duads

thus implies thatGis transitive on triples. Moreover, by3.6Gδhas just three orbits on duads

which must consist of{a, b}itself, of length 1, the 2n−2duads such as{a, c}which intersect

δin one point, and then−2₂ duads disjoint fromδ.

We now wish to show thatGacts triply transitively onΩ. So let{a, b, c}and{x, y, z}

be two triples of points inΩ; we must produce a permutationπ ∈Gsuch thatxπ a, yπ b, zπ _c_{. Now by the transitivity on duads, there exists a} _σ

1 ∈ G with {x, y}σ1 {a, b}.

So{x, z}σ1 _{is a duad intersecting}_δ_{in 1 point. But}_G

δ acts transitively on the set of 2n−2

such duads, and so there exists a σ2 ∈ Gδ such that {xσ1, zσ1}σ2 {a, c} {x, z}σ1σ2 and {x, y}σ1σ2 {_{a, b}}_{. Thus,}_xσ1σ2 _{a, y}σ1σ2 _b_and_zσ1σ2 _c_{as required.}

That G acts transitively on tetrads, that is, that G is 4-homogeneous, follows immediately from the fact thatGis transitive on duads, andGδis transitive on duads disjoint

fromδ. We now investigate from the first principles the circumstances in whichGis not 4-transitive.

LetT {a, b, c, d}denote a tetrad, then theGT denotes thesetstabilizer ofT inG,

andG_Tdenotes the point-wise stabilizer ofT. The factor groupFGT/GTgives the action

ofGTon the four points ofTand is thus isomorphic to a subgroup of S4. IfF ∼S4, thenGis

4-transitive, since any tetrad can be mapped intoT, and then the four points{a, b, c, d}may be rearranged as we wish by elements ofGT. We shall now show that ifFS4, thenF ∼A4.

Ifδ1 andδ2are two disjoint duads, then the transitivity ofGon duads together with

the transitivity ofGδ1on duads disjoint fromδ1show that

|G|

n

2

n−2

2

|L| nn−1n−2n−3

4 |L|, 3.7

whereLdenotes the stabilizer ofδ1andδ2. Then,Ga, the stabilizer of a point, has, order

|Ga| n−1n−2n−3

4 |L|, 3.8

which is visibly divisible by 3, and soGcontains elements of order 3 which fix a point. Thus, the factor group F contains elements of cycle shape 1.3. Now by the triple transitivity, we see that the point stabilizerGaacts double transitively onΩ\ {a}and thus has even order.

So G contains involutions of cycle shape 1r_.₂s _where _r _≥ _{1 and}_s _≥ _{2, for if} _s _{1, then}

Gcontains transpositions, and the double transitivity ofGwill then force G ∼ Sn. Thus,F

contains elements which act as 22_{on the four points of}_T_{. The only subgroups of S}

4 which

contain elements of order 3 and such involutions are A4and S4.

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assume that every involution ofGhas just one fixed point. Moreover, no element ofGmay contain a 4-cycle as we could conjugate this 4-cycle into the positions ofT and would again haveF ∼ S4. Thus every nontrivial element in a Sylow 2-subgroup ofGhas order 2 and so

Sylow 2-subgroups ofGare elementary abelian.

We have seen that the 2-point stabilizerGabinGis of odd order,and, therefore, by the

Feit-Thompson theorem, it is soluble. A minimal normal subgroup of a soluble group is an elementary abelianp-group, see, for example, Scott13, page 74, where in this casepis odd. IfK ≤ Gab is such a subgroup, then, sinceGab acts transitively on the duads disjoint from {a, b},K must act transitively onΩ\ {a, b}, otherwise duads in diﬀerent orbits ofKcould not be mapped to duads in the same orbit. Moreover,Kis abelian and so by Lemma2.10K

acts regularly. Any other minimal normal subgroupLwould have to intersectKtrivially as the intersection will also be normal. But ifl ∈ L, k∈K thenl, k l−1k−1lk∈L∩K 1, and solcommutes with the transitive subgroupK. ThusL, Kis abelian and transitive, and so regular. Thus,L, KKL, andKis unique.|K|pd _n₋_2.

We now investigate the structure of the point stabilizerGaand will in fact show that

it has a regular normal subgroup which is an elementary abelian 2-group. At this point, our work is close to that of Kantor9, and we include, part of his argument: letτ ∈ G{a,b} be a permutation which interchangesaand b; sinceτ2 _∈ _G

ab, a group of odd order, we may

assume thatτ is an involution. By the uniqueness ofKwe see thatKτ _K_{and so}_|_{K, τ}_|

2|K| 2n−2. Moreover,τ has a unique fixed point, andKacts transitively onΩ\ {a, b}, and so|τK_|_n₋_{2. Thus, we have}

τK{τ1τ, τ2, . . . , τn−2}Kτ, 3.9

where eachτi is an involution interchangingaand b. SoK {τ1τ, τ2τ, . . . , τn−2τ}, and it is

clear that conjugation byτinverts every element ofK. But now we can prove the following.

Lemma 3.4. No two involutions ofGwhich contain the same transposition can fix the same point, nor can they have a further transposition in common.

Proof. We may assume that the involutionsσ1 andσ2 both contain the transpositiona, b.

Then bothσ1andσ2 invert every element ofK, and soσ1σ2 commutes with every element

ofK, and fixesaandb. Ifσ1andσ2 either have a fixed point in common or have a further

transposition in common, thenσ1σ2 fixes a point and commutes with a transitive groupon Ω\ {a, b}. Thus,σ1σ21, and soσ1 σ2.

Corollary 3.5. For any tetradT {a, b, c, d}there is a unique involution acting asa bc don

the points ofTand a unique Klein four-group acting onT.

Proof. We have seen that the factor groupF acting on the four points ofT is isomorphic to

A4; by the Lemma the elementπ1 acting asa bc dis unique. But if we conjugate this

element by the elementπ2 which acts asa cb d, we wil obtain an involution acting as a bc donT which can only beπ1. Thus,π₁π2 π1 andπ1π2 π2π1 π3, the unique

involution which acts asa db conT.

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we see that there is a unique involution fixing a given point and containing a given trans-position; thus, the number of involutions fixing a given point is

n−1

2

_n₋₁

2 n−2. 3.10

It is clear that commuting involutions must fix one another’s set of fixed points, and so, since our involutions have just one fixed point, commuting involutions have the same fixed points. Now a given involution is in

⎛ ⎝n−₂1

2 ⎞

⎠ n−1n−3

8 3.11

Klein four-group, and each pair of commuting involutions fixesn−1/4 tetrads. Thus, the number of involutions commuting with a given involution is

n−1n−3

8 ×2÷

n−1

4 n−3. 3.12

Thus then−2involutions which fix a given point all commute with one another and so must form the nontrivial elements of an elementary abelian 2-group which must be normal in the point stabilizer. Thus,n2m_{1 for some}_m_≥_{2, and 2}m₋₁_pd_{for some}_d_where_p_is

an odd prime. We have the following.

Lemma 3.6. If 2m_pd_{1 where}_p_{is prime, then}_d_{1 and}_m_{is prime.}

Proof. Ifp≡1 mod 4, the right-hand side is congruent to 2 modulo 4 which is a contradiction

sincem ≥ 2, and so the left-hand side is congruent to 0 modulo 4. So p ≡ 3 mod 4, and

dmust be odd, since evendwould result in the left-hand side being again congruent to 2 modulo 4. But now

2mpd1p1pd−1− · · · −p1

dterms

. 3.13

The underbracketed term is a sum of an odd number of odd integers and so is odd, which is a contradiction unlessd1. But then 2m₋_{1 cannot be prime unless}_m_{is prime.}

The point stabilizing subgroupGathus has the shape

Ga∼2m:P :H, 3.14

wherePis a cyclic subgroup of orderp2m₋_{1 and}_H_normalizes_P_{. If}_P _x_{, then}_x_acts

on the elementary abelian group of order 2m_{, which may be regarded as an}_m_-dimensional

vector space overZ2, as a cycle of maximal length—known as a Singer cycle. The subgroup

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appeal to Theorem 7.3 in Huppert11, page 187, which for convenience we have included here as Theorem2.11. For our purposes, we havef 1 andp2; the Theorem says that our groupP : His a subgroup of a Frobenius group of shapep : m, so eitherH ∼ Cm orHis

trivial asmis prime. We thus have

|G| 2m12m2m−1k, 3.15

wherekmor 1,mis prime, andn2m_1.

ButGis transitive on tetrads, and the stabilizer of a tetrad acts as A4on the 4 points of

the tetrad. Thus,n

4×12 divides|G|, and we have

nn−1n−2k nn−1n−2n−3

4! .12.s, 3.16

for some integers; son−3/2 dividesk. Ifk 1, we must haven 5, whenGis a triply transitive subgroup of S5of the order 5.4.3 60. ThusG ∼ A5. Ifk m, we have 2m−1−1

dividesm, and sincemis prime we have 2m−1m1, and som3. In this case,|G|9.8.7.3. We now construct a permutation group of this order and degree 9 and show that it is unique. We shall take the set Ω to be Ω {a , b ∞,0,1, . . . ,6}, that is, to say the projective line P17 supplemented by the symbol . Without loss of generality, we take

x 0 1 2 3 4 5 6normalized withinGa G byy 1 2 43 6 5. We now seek

an elementary abelian groupEof order 8 which is normalized byx, y ∼7 : 3. Now one of the 7 involutions ofEmust commute withy, and since it has cycle shape 1.24, it must be one of

i σ1 ∞ 01 32 64 5,

ii σ2 ∞ 01 52 34 6,

iii σ3 ∞ 01 62 54 3.

3.17

Bothσ₁x,yandσ₂x,yare indeed elementary abelian groups of order 8; however,σx₃ does not commute withσ3. Moreover, the the permutation∞01 62 53 4of the symmetric

group S8, which negates the elements ofZ7, normalizesx, yand interchangesσ1andσ2. So

without loss of generality, we may assume thatGx, y, σ1 ∼23: 7 : 3.

We now consider the number of Sylow 7-subgroups which G contains. The only permutation of S9outsidePwhich commutes withPis the transposition ∞which does

not have the desired cycle shapeand, in any case, would generate S9. Thus,CGP P. We know thatNGP x, y ∼ 7 : 3 which has the index 9.8.7.3/7.3 72 in G. Now

72≡2 modulo7, and soNGPmust have the order 42 and contain an involution commuting withyand invertingx. This element must have, cycle shape 1.24_{as usual, and so without loss}

of generality it isz ∞1 62 53 4. Since generators forGwere reached without loss of generality, and since the group PΣL28—which consists of the projective special linear group PSL28with the field automorphism adjoined—is well known to have the required properties, we must have

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[image:13.600.99.506.109.153.2]

Table 4: The projective plane P18.

ηi _∞ ₀ ₁ _η _η2 _η3 _η4 _η5 _η6

fη ∞ 0 1 η η2 _η₁ _η2_η _η2_η₁ _η2₁

Ω ∞ 0 1 2 3 4 5 6

However, for completeness, we exhibit the identification as permutations of the projective line P18. The field of order 8 is obtained by extendingZ2by a root ofη3_η_{1. In Table}₄

we display elements of P18both as powers ofηand as quadratic expressions inη. We also show the correspondence with elements ofΩas used above. Our generators now become

x : u−→ηu, y : u−→u2, σ1 : u−→u1,

z : u−→1/u.

3.19

This completes the proof of the main theorem.

References

1 T. Tsuzuku, “On multiple transitivity of permutation groups,” Nagoya Mathematical Journal, vol. 18, pp. 93–109, 1961.

2 J. Saxl, “Characters of multiply transitive permutation groups,” Journal of Algebra, vol. 34, pp. 528–539, 1975.

3 P. M. Neumann, “Generosity and characters of multiply transitive permutation groups,” Proceedings

of the London Mathematical Society, vol. 31, no. 4, pp. 457–481, 1975.

4 J. Saxl, “The complex characters of the symmetric groups that remain irreducible in subgroups,”

Journal of Algebra, vol. 111, no. 1, pp. 210–219, 1987.

5 J. Saxl, “Characters and generosity of permutation groups,” Proceedings of the American Mathematical

Society, vol. 47, pp. 73–76, 1975.

6 G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University Press, New York, NY, USA, 2nd edition, 2001.

7 G. D. James, The Representation Theory of the Symmetric Groups, vol. 682 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978.

8 W. M. Kantor, “4-homogeneous groups,” Mathematische Zeitschrift, vol. 103, pp. 67–68, 1968.

9 W. M. Kantor, “k-homogeneous groups,” Mathematische Zeitschrift, vol. 124, pp. 261–265, 1972.

10 D. Livingstone and A. Wagner, “Transitivity of finite permutation groups on unordered sets,”

Mathematische Zeitschrift, vol. 90, pp. 393–403, 1965.

11 B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1967.

12 S. Aldhafeeri, Characters of multiply transitive groups, Ph.D. thesis, University of Birmingham, 2009.

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