Let G be a permutation group of degree n viewed as a subgroup of the symmetric group S≅S_n. We show that if the irreducible character of S corresponding to the partition of n into subsets of sizes n − 2 and 2, that is, to say the character often denoted by χ^(n−2,2), remains irreducible when restricted to G, then n = 4, 5 or 9 and G≅S₃, A₅, or PΣL₂(8), respectively, or G is 4-transitive.

1. Introduction

Let G be a permutation group of degree n. The connection between the multiple transitivity of G and the irreducibility of certain irreducible characters of of the symmetric group S_n when restricted to G goes back to Frobenius, and a proof of his classical result can be found in Tsuzuku [1]. In [2], Saxl strengthens this result, limiting the irreducibility hypothesis to the characters , a special case of his theorem having been proved by Neumann [3]. In [4] Saxl uses the Classification of Finite Simple Groups to go much further. He obtains a complete list of pairs (χ, G) where χ is an irreducible character of the symmetric group S_n, and G is a subgroup of S_n such that the restriction of χ to G remains irreducible.

The present paper is concerned with the case when 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 Theorem 3.3, can be deduced from [2, 3, 5]. However, the approach used here is self-contained and elementary.

2. Notation and Preliminary Results

2.1. Permutation Characters

We start by recalling two fundamental results from the elementary permutation character theory. If a group G permutes a set X = {1,2, …, n} then, we let B = {e₁, e₂, …, e_n} be a basis of an n-dimensional complex vector space, which becomes a ℂG-module when we define e_iν = e_ν(i), for ν ∈ G, and extend linearly. If π_X denotes the character afforded by this representation and the permutation ν has cycle shape 1^r.2^s… when written as a product of disjoint cycles, then

(2.1)

where Fix _X(ν) denotes the set of points of X fixed by the permutation ν. The subscript X will be omitted when there is no fear of confusion. The standard inner product defined on the space of class functions when restricted to (generalized) characters becomes

(2.2)

where χ and ψ are characters of the group G, and

denotes the complex conjugate of the complex number α. The trivial character of G which takes the value 1 on every element of G will be denoted by 1_G. 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. If G is a permutation group acting on the set X with permutation character π_X, then we have the following result:

(2.3)

where orb(G, X) denotes the number of orbits when G acts on X.

Proof. Firstly note that

(2.4)

where Stab_G(a) denotes the stabilizer in G of a. So, if G has orbits O₁, O₂, …, O_m when it acts on X, the Orbit-Stabilizer theorem implies that

(2.5)

where b_i is a representative of the orbit O_i. Thus,

(2.6)

If the group G acts on the set X and on the set Y with permutations characters π_X and π_Y, respectively, then G also acts on the Cartesian product X × Y by defining

(2.7)

Theorem 2.2. If the group G acts on the sets X and Y with permutation characters π_X and π_Y respectively, then

(2.8)

Proof.

(2.9)

Corollary 2.3. If G acts transitively on X and a ∈ X, then 〈π_X, π_Y〉 = orb(G_a, Y), the number of orbits of the stabilizer of a point in X on Y.

Corollary 2.4. If G acts transitively on X and a ∈ X, then 〈π_X, π_X〉 = orb(G_a, X), the rank of the permutation action of G on X.

In the case when S≅S_n, the symmetric group acting on an n-element set Ω = {1,2, …, n}, we denote the associated permutation character by π₁. More generally π_k will denote the permutation action of S≅S_n on unordered k-element subsets. In particular, π₂ will denote the permutation character of the action of S on Δ = {{a, b}∣a, b ∈ Ω}, the set of unordered pairs of elements of Ω which we shall call duads. Thus, if ν ∈ S has cycle shape 1^r.2^s…, then

(2.10)

By Theorem 2.1 and its corollaries we see that

(2.11)

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, δ say, the orbit containing duads which intersect δ in one point, and the orbit of duads disjoint from δ. We thus have that

(2.12)

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

2.2. The Irreducible Characters of the Symmetric Group S_n

Definition 2.5. A partition λ of the integer n is a sequence λ = (λ₁, λ₂, …, λ_k) where λ_i ≥ λ_i+1 for i = 1, …, k − 1 and .

The irreducible characters of S_n are in one-to-one correspondence with the partitions of n and we denote the character corresponding to λ by χ^λ. In this notation, the trivial character of S_n corresponds to the partition of n into a single subset of size n and is thus denoted by 1_S = χ⁽ⁿ⁾. The permutation character of degree n is given by π₁ = χ⁽ⁿ⁾ + χ^(n−1,1). Indeed we have the general result.

Theorem 2.6. If k ≤ n/2 and π_k denotes the permutation character of the action of S_n on unordered k-subsets, then

(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 Liebeck [6, page 344]. That these irreducibles are indeed those corresponding to the above partitions may be verified using the celebrated Murnaghan-Nakayama rule, see [7, page 79], which enables one to evaluate the character value of the irreducible χλ on any conjugacy class of the symmetric group. We see that for k ≤ n/2 we have

(2.14)

and so, if σ ∈ S_n, then

(2.15)

In particular, the degree of χ^(n−k,k) is given by

(2.16)

Applying (2.15) to a permutation σ with cycle shape 1^r.2^s…, we have

(2.17)

2.3. Multiple Transitivity

Definition 2.7. The group G is said to act k-transitively on the set X if, and only if, given (x₁, x₂, …, x_k) and (y₁, y₂, …, x_k), two ordered k-tuples of distinct points of X, there exists a permutation σ ∈ G such that for i = 1 ⋯ k.

Definition 2.8. The group G is said to act k-homogeneously on the set X if, and only if, given A, B ⊂ X with |A| = |B| = k, there exists a permutation σ ∈ G such that A^σ = B.

In other words, G is k-homogeneous if it acts transitively on the k-subsets. Those fascinating groups which are k-homogeneous without being k-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 S_n on 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 that G has even order and acts 2-homogeneously on X. Let α = {a₁, a₂} and β = {b₁, b₂} be two duads; we must produce a permutation which maps a_i to b_i for i = 1 and 2. By 2-homogeneity, there exists a σ ∈ G such that α^σ = β, and since G is even and again using the 2-homogeneity, there exists an element τ ∈ G of order 2 which acts as τ = (b₁ b₂)⋯. If , then we are done, otherwise στ will suffice.

2.4. Restriction to a Subgroup of S_n

In what follows, the subgroup G will be a fixed subgroup of S≅S_n, and we shall denote the restriction of the character χ of S to G by

(2.18)

In order to simplify the notation, we shall denote

(2.19)

So we shall have

(2.20)

The restriction to G of an irreducible character of S may or may not remain irreducible. However, if G ≤ S≅S_n, then using Theorems 2.1 and 2.2 we have

(1)
〈ψ₁, 1_G〉 is the number of orbits of G on points,
(2)
〈ψ₂, 1_G〉 is the number of orbits of G on duads,
(3)
〈ψ₁, ψ₁〉 is the number of orbits of G on Ω × Ω,
(4)
〈ψ₁, ψ₂〉 is the number of orbits of G on Ω × Δ, and
(5)
〈ψ₂, ψ₂〉 is the number of orbits of G on Δ × Δ.

Note that if χ and ψ are as in (2.12), then χ|_G = ψ^(n−1,1) and ψ|_G = ψ^(n−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. Let K be a transitive subgroup of the symmetric group S_n acting on the set Ω, and let the permutation σ ∈ S_n commute with K and fix a point a ∈ Ω. Then F = Fix(σ) ⊂ Ω is invariant under the action of K and is thus the empty set ϕ or Ω. But a ∈ F ≠ ϕ, and so F = Ω and σ = 1.

If K is abelian and σ ∈ K_a, the stabilizer in K of a, then σ commutes with K and fixes the point a. Thus, σ = 1, from the above, and so K_a = 〈1〉, and the transitive group K acts 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 GL_m(p^f) contains a cyclic subgroup K of order p^mf − 1. We have C_G(K) = K and that N_G(K)/K is cyclic of order m. (Such cycles of maximal possible length are referred to as Singer cycles.)

3. Statement and Proof of Main Theorem

Let G ≤ S_n. The transitivity of G is related to the reducibility of certain irreducible characters of S upon restriction to G as follows.

Lemma 3.1. G is doubly transitive ⇔ψ^(n−1,1) ∈ Irr(G).

Proof. Follows immediately from Corollary 2.4.

Lemma 3.2. G is 4-transitive, n ≥ 4⇒ψ^(n−1,1), ψ^(n−2,2) ∈ Irr(G).

Proof. If G is 4-transitive, then it is certainly doubly transitive, and so ψ^(n−1,1) ∈ Irr(G). Moreover, if δ is a duad, then G_δ certainly has just three orbits on duads, as described towards the end of Section 2.1. Thus,

(3.1)

and so 〈ψ^(n−2,2), ψ^(n−2,2)〉 = 1 and ψ^(n−2,2) ∈ Irr(G).

It is thus natural to ask in what circumstances we can have ψ^(n−2,2) ∈ Irr(G) without 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 Kantor [8] but is otherwise self-contained and based on Aldhafeeri [12].

Theorem 3.3. Let G be a subgroup of the symmetric group of degree n with n ≥ 4, thus G ≤ S≅S_n, and let χ^(n−2,2) represent the irreducible character of S corresponding to the partition of n into a pair and a subset of cardinality n − 2. Then

(3.2)

(i.e., the restriction of χ^(n−2,2) to G is an irreducible representation of G) if, and only if,

(i)
n = 4 and G≅S₃,
(ii)
n = 5 and G≅A₅,
(iii)
n = 9 and G≅PΣL₂(8), or
(iv)
G is 4-transitive on n letters.

Proof. Note that the character χ^(n−2,2) of S_n only exists for n ≥ 4, and so the restriction on n is only required to make the theorem meaningful. In case (i), the subgroup G is not transitive, but no restriction was placed on the transitivity of G.

We shall assume that G is not 4-transitive and that ψ^(n−2,2) ∈ Irr(G). We have ψ^(n−2,2)(1) = n(n − 3)/2 ≥ 2 for n ≥ 4. Thus,

(3.3)

and so orb(G, Ω) = orb(G, Δ) = k, say. But if orb(G, Ω) = k then orb

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

ways. Now

for k > 3, and if k = 3, at least one of the three orbits must have more than one point in it (since n ≥ 4), and so orb

. Thus, k ≤ 2. If k = 2 and both orbits are of cardinality greater than 1 then orb(G, Δ) ≥ 3, a contradiction, so we must have one orbit of length 1 and the other of length n − 1.

Now ψ^(n−2,2)(1) = n(n − 3)/2 > n − 1 = ψ^(n−1,1)(1) provided n² − 5n + 2 > 0, which holds for n ≥ 5. So provided n > 4, we have 〈ψ^(n−2,2), ψ^(n−1,1)〉 = 0. Thus,

(3.4)

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 S₄ are A₄, D₈, and S₃, fixing a point. In the following diagram we show the restrictions of χ^(2,2) to subgroups of S₄ isomorphic to A₄, D₈, and S₃, respectively. The top line in Tables 1, 2, and 3 gives 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 class (viewed 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.

A₄	12	4	3	3
A₄	1⁴	2²	1.3	1.3
ψ^(2,2)	2	2	−1	−1
	1	1	ω
	1	1		ω

D₈	8	8	4	4	4
D₈	1⁴	2²	4	2²	1².2
ψ^(2,2)	2	2	0	0	2
	1	1	1	1	1
	1	1	−1	−1	1

S₃	6	3	2
S₃	1⁴	1.3	1².2
ψ^(2,2)	2	−1	0
ψ₁	4	1	2

We may now assume that n ≥ 5 and that (3.4) holds. Let us return to the case reached above where G has two orbits on Ω, of lengths 1 and n − 1, respectively. In order to dismiss this case (for n > 4), we may assume that G≅S_n−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 of G are {a} and {b₁, b₂, …, b_n−1}. But now orb(G, Ω × Ω) = 5 with orbit representatives {(a, a), (a, b₁), (b₁, a), (b₁, b₁), (b₁, b₂)}. However, orb(G, Ω × Δ) = 6 with orbit representatives

(3.5)

(Note that when n = 4 and G≅S₃, we do indeed have 〈ψ₁, ψ₁〉 = 5 and that 〈ψ₁, ψ₂〉 = 〈ψ₁, ψ₁ + ψ^(2,2)〉 = 6). However, if n > 4, this contradicts (3.4), and so we must have that G acts transitively on Ω and on Δ.

Now let Ω = {a, b, c, …}. The transitivity of G together with (3.4) imply that the stabilizer of a point, G_a, has the same number of orbits on points as it has on duads. As above, if G_a has k orbits on Ω, then it has at least

orbits on duads, so we must have that 2 ≤ k ≤ 3. If k = 3, then at least one of the orbits of G_a must have length greater than 1 (since n ≥ 5), and so the number of orbits of G_a on duads is at least 4, a contradiction. Thus, k = 2 and G is doubly transitive. In particular, ψ^(n−1,1), ψ^(n−2,2) ∈ Irr(G), and we have

(3.6)

Since G_δ, the stabilizer in G of the duad δ = {a, b}, has just two orbits on Ω, these must be {a, b} and {c, d, …}, and so G_δ acts transitively on Ω∖{a, b}. The transitivity of G on duads thus implies that G is transitive on triples. Moreover, by (3.6) G_δ has just three orbits on duads which must consist of {a, b} itself, of length 1, the 2(n − 2) duads such as {a, c} which intersect δ in one point, and the

duads disjoint from δ.

We now wish to show that G acts triply transitively on Ω. So let {a, b, c} and {x, y, z} be two triples of points in Ω; we must produce a permutation π ∈ G such that x^π = a, y^π = b, z^π = c. Now by the transitivity on duads, there exists a σ₁ ∈ G with . So is a duad intersecting δ in 1 point. But G_δ acts transitively on the set of 2(n − 2) such duads, and so there exists a σ₂ ∈ G_δ such that and . Thus, and as required.

That G acts transitively on tetrads, that is, that G is 4-homogeneous, follows immediately from the fact that G is transitive on duads, and G_δ is transitive on duads disjoint from δ. We now investigate from the first principles the circumstances in which G is not 4-transitive.

Let T = {a, b, c, d} denote a tetrad, then the G_T denotes the (set) stabilizer of T in G, and G_[T] denotes the point-wise stabilizer of T. The factor group F = G_T/G_[T] gives the action of G_T on the four points of T and is thus isomorphic to a subgroup of S₄. If F≅S₄, then G is 4-transitive, since any tetrad can be mapped into T, and then the four points {a, b, c, d} may be rearranged as we wish by elements of G_T. We shall now show that if F≇S₄, then F≅A₄.

If δ₁ and δ₂ are two disjoint duads, then the transitivity of G on duads together with the transitivity of

on duads disjoint from δ₁ show that

(3.7)

where L denotes the stabilizer of δ₁ and δ₂. Then, G_a, the stabilizer of a point, has, order

(3.8)

which is visibly divisible by 3, and so G contains 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 stabilizer G_a acts double transitively on Ω∖{a} and thus has even order. So G contains involutions of cycle shape 1^r.2^s where r ≥ 1 and s ≥ 2, for if s = 1, then G contains transpositions, and the double transitivity of G will then force G≅S_n. Thus, F contains elements which act as 2² on the four points of T. The only subgroups of S₄ which contain elements of order 3 and such involutions are A₄ and S₄.

Note further that if G contains involutions which fix more than 1 point, then F will contain transpositions; we shall then have F≅S₄, and G will be 4-transitive. So we may assume that every involution of G has just one fixed point. Moreover, no element of G may contain a 4-cycle as we could conjugate this 4-cycle into the positions of T and would again have F≅S₄. Thus every nontrivial element in a Sylow 2-subgroup of G has order 2 and so Sylow 2-subgroups of G are elementary abelian.

We have seen that the 2-point stabilizer G_ab in G is of odd order,and, therefore, by the Feit-Thompson theorem, it is soluble. A minimal normal subgroup of a soluble group is an elementary abelian p-group, see, for example, Scott [13, page 74], where in this case p is odd. If K ≤ G_ab is such a subgroup, then, since G_ab acts transitively on the duads disjoint from {a, b}, K must act transitively on Ω∖{a, b}, otherwise duads in different orbits of K could not be mapped to duads in the same orbit. Moreover, K is abelian and so by Lemma 2.10 K acts regularly. Any other minimal normal subgroup L would have to intersect K trivially as the intersection will also be normal. But if l ∈ L, k ∈ K then [l, k] = l⁻¹k⁻¹lk ∈ L∩K = 〈1〉, and so l commutes with the transitive subgroup K. Thus 〈L, K〉 is abelian and transitive, and so regular. Thus, 〈L, K〉 = K = L, and K is unique. |K| = p^d = n − 2.

We now investigate the structure of the point stabilizer G_a and 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 Kantor [9], and we include, part of his argument: let τ ∈ G^{a,b} be a permutation which interchanges a and b; since τ² ∈ G_ab, a group of odd order, we may assume that τ is an involution. By the uniqueness of K we see that K^τ = K and so |〈K, τ〉| = 2|K| = 2(n − 2). Moreover, τ has a unique fixed point, and K acts transitively on Ω∖{a, b}, and so |τ^K| = n − 2. Thus, we have

(3.9)

where each τ_i is an involution interchanging a and b. So K = {τ₁τ, τ₂τ, …, τ_n−2τ}, and it is clear that conjugation by τ inverts every element of K. But now we can prove the following.

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

Proof. We may assume that the involutions σ₁ and σ₂ both contain the transposition (a, b). Then both σ₁ and σ₂ invert every element of K, and so σ₁σ₂ commutes with every element of K, and fixes a and b. If σ₁ and σ₂ either have a fixed point in common or have a further transposition in common, then σ₁σ₂ fixes a point and commutes with a transitive group (on Ω∖{a, b}). Thus, σ₁σ₂ = 1, and so σ₁ = σ₂.

Corollary 3.5. For any tetrad T = {a, b, c, d} there is a unique involution acting as (a b)(c d) on the points of T and a unique Klein four-group acting on T.

Proof. We have seen that the factor group F acting on the four points of T is isomorphic to A₄; by the Lemma the element π₁ acting as (a b)(c d) is unique. But if we conjugate this element by the element π₂ which acts as (a c)(b d), we wil obtain an involution acting as (a b)(c d) on T which can only be π₁. Thus, and π₁π₂ = π₂π₁ = π₃, the unique involution which acts as (a d)(b c) on T.

We are now in a position to count the number of involutions in G and, in particular, to count the number of involutions which commute with a given involution. From Lemma 3.4, we see that there is a unique involution fixing a given point and containing a given transposition; thus, the number of involutions fixing a given point is

(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

(3.11)

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

(3.12)

Thus the (n − 2) involutions 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, n = 2^m + 1 for some m ≥ 2, and 2^m − 1 = p^d for some d where p is an odd prime. We have the following.

Lemma 3.6. If 2^m = p^d + 1 where p is prime, then d = 1 and m is prime.

Proof. If p ≡ 1 mod 4, the right-hand side is congruent to 2 modulo 4 which is a contradiction since m ≥ 2, and so the left-hand side is congruent to 0 modulo 4. So p ≡ 3 mod 4, and d must be odd, since even d would result in the left-hand side being again congruent to 2 modulo 4. But now

(3.13)

The underbracketed term is a sum of an odd number of odd integers and so is odd, which is a contradiction unless d = 1. But then 2^m − 1 cannot be prime unless m is prime.

The point stabilizing subgroup G_a thus has the shape

(3.14)

where P is a cyclic subgroup of order p = 2^m − 1 and H normalizes P. If P = 〈x〉, then x acts on the elementary abelian group of order 2^m, which may be regarded as an m-dimensional vector space over ℤ₂, as a cycle of maximal length—known as a Singer cycle. The subgroup P : H is thus acting faithfully as a subgroup of the general linear group GL_m(2). We now appeal to Theorem 7.3 in Huppert [11, page 187], which for convenience we have included here as Theorem 2.11. For our purposes, we have f = 1 and p = 2; the Theorem says that our group P : H is a subgroup of a Frobenius group of shape p : m, so either H≅C_m or H is trivial as m is prime. We thus have

(3.15)

where k = m or 1, m is prime, and n = 2^m + 1.

But G is transitive on tetrads, and the stabilizer of a tetrad acts as A₄ on the 4 points of the tetrad. Thus, divides |G|, and we have

(3.16)

for some integer s; so (n − 3)/2 divides k. If k = 1, we must have n = 5, when G is a triply transitive subgroup of S₅ of the order 5.4.3 = 60. Thus G≅A₅. If k = m, we have 2^m−1 − 1 divides m, and since m is prime we have 2^m−1 = m + 1, and so m = 3. 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 P₁(7) supplemented by the symbol ⋆. Without loss of generality, we take x = (0 1 2 3 4 5 6) normalized within G_a = G_⋆ by y = (1 2 4)(3 6 5). We now seek an elementary abelian group E of order 8 which is normalized by 〈x, y〉≅7 : 3. Now one of the 7 involutions of E must commute with y, and since it has cycle shape 1.2⁴, it must be one of

(3.17)

Both

and

are indeed elementary abelian groups of order 8; however,

does not commute with σ₃. Moreover, the the permutation (∞)(0)(1 6)(2 5)(3 4) of the symmetric group S₈, which negates the elements of ℤ₇, normalizes 〈x, y〉 and interchanges σ₁ and σ₂. So without loss of generality, we may assume that G_⋆ = 〈x, y, σ₁〉≅2³ : 7 : 3.

We now consider the number of Sylow 7-subgroups which G contains. The only permutation of S₉ outside P which commutes with P is the transposition (⋆ ∞) which does not have the desired cycle shape (and, in any case, would generate S₉). Thus, C_G(P) = P. We know that which has the index 9.8.7.3/7.3 = 72 in G. Now 72 ≡ 2 modulo7, and so N_G(P) must have the order 42 and contain an involution commuting with y and inverting x. This element must have, cycle shape 1.2⁴ as usual, and so without loss of generality it is z = (⋆ ∞)(1 6)(2 5)(3 4). Since generators for G were reached without loss of generality, and since the group PΣL₂(8)—which consists of the projective special linear group PSL₂(8) with the field automorphism adjoined—is well known to have the required properties, we must have

(3.18)

However, for completeness, we exhibit the identification as permutations of the projective line P₁(8). The field of order 8 is obtained by extending ℤ₂ by a root of η³ + η + 1. In Table 4 we display elements of P₁(8) both as powers of η and as quadratic expressions in η. We also show the correspondence with elements of Ω as used above. Our generators now become

(3.19)

This completes the proof of the main theorem.

Table 4. The projective plane P₁(8).

ηⁱ	∞	0	1	η	η²	η³	η⁴	η⁵	η⁶
f(η)	∞	0	1	η	η²	η + 1	η² + η	η² + η + 1	η² + 1
Ω	⋆	∞	0	1	2	3	4	5	6

References

1 Tsuzuku T., On multiple transitivity of permutation groups, Nagoya Mathematical Journal. (1961) 18, 93–109, 0124420, ZBL0109.01503.
10.1017/S0027763000002269
Google Scholar
2 Saxl J., Characters of multiply transitive permutation groups, Journal of Algebra. (1975) 34, 528–539, 0450385, https://doi.org/10.1016/0021-8693(75)90174-X, ZBL0323.20004.
10.1016/0021-8693(75)90174-X
Web of Science® Google Scholar
3 Neumann P. M., Generosity and characters of multiply transitive permutation groups, Proceedings of the London Mathematical Society. (1975) 31, no. 4, 457–481, 0382407, https://doi.org/10.1112/plms/s3-31.4.457, ZBL0324.20003.
10.1112/plms/s3-31.4.457
Web of Science® Google Scholar
4 Saxl J., The complex characters of the symmetric groups that remain irreducible in subgroups, Journal of Algebra. (1987) 111, no. 1, 210–219, 913206, https://doi.org/10.1016/0021-8693(87)90251-1, ZBL0633.20008.
10.1016/0021-8693(87)90251-1
Web of Science® Google Scholar
5 Saxl J., Characters and generosity of permutation groups, Proceedings of the American Mathematical Society. (1975) 47, 73–76, 0357563, https://doi.org/10.1090/S0002-9939-1975-0357563-X, ZBL0264.20005.
10.1090/S0002-9939-1975-0357563-X
Web of Science® Google Scholar
6 James G. and Liebeck M., Representations and Characters of Groups, 2001, 2nd edition, Cambridge University Press, New York, NY, USA, 1864147, ZBL1210.74023.
10.1017/CBO9780511814532
Google Scholar
7 James G. D., The Representation Theory of the Symmetric Groups, 1978, 682, Springer, Berlin, Germany, Lecture Notes in Mathematics, 513828, ZBL0499.10032.
10.1007/BFb0067708
Google Scholar
8 Kantor W. M., 4-homogeneous groups, Mathematische Zeitschrift. (1968) 103, 67–68, 0220812, https://doi.org/10.1007/BF01111290, ZBL0189.02303.
10.1007/BF01111290
Web of Science® Google Scholar
9 Kantor W. M., k-homogeneous groups, Mathematische Zeitschrift. (1972) 124, 261–265, 0306296, https://doi.org/10.1007/BF01113919, ZBL0232.20003.
10.1007/BF01113919
Web of Science® Google Scholar
10 Livingstone D. and Wagner A., Transitivity of finite permutation groups on unordered sets, Mathematische Zeitschrift. (1965) 90, 393–403, 0186725, https://doi.org/10.1007/BF01112361, ZBL0136.28101.
10.1007/BF01112361
Web of Science® Google Scholar
11 Huppert B., Endliche Gruppen. I, 1967, Springer, Berlin, Germany, Die Grundlehren der Mathematischen Wissenschaften, 0224703, ZBL0217.07201.
10.1007/978-3-642-64981-3
Google Scholar
12 Aldhafeeri S., Characters of multiply transitive groups, Ph.D. thesis, 2009, University of Birmingham.
Google Scholar
13 Scott W. R., Group Theory, 1964, Prentice-Hall, 0167513.
Google Scholar

All articles

A Character Condition for Quadruple Transitivity

Abstract

1. Introduction

2. Notation and Preliminary Results

2.1. Permutation Characters

2.2. The Irreducible Characters of the Symmetric Group S_n

2.3. Multiple Transitivity

2.4. Restriction to a Subgroup of S_n

2.5. General Preliminary Results

3. Statement and Proof of Main Theorem

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

A Character Condition for Quadruple Transitivity

Abstract

1. Introduction

2. Notation and Preliminary Results

2.1. Permutation Characters

2.2. The Irreducible Characters of the Symmetric Group Sn

2.3. Multiple Transitivity

2.4. Restriction to a Subgroup of Sn

2.5. General Preliminary Results

3. Statement and Proof of Main Theorem

References

References

Related

Information

2.2. The Irreducible Characters of the Symmetric Group S_n

2.4. Restriction to a Subgroup of S_n