Coprime subdegrees for primitive permutation groups and completely reducible linear groups

Dolfi, Silvio; Guralnick, Robert M.; Praeger, Cheryl E.; Spiga, Pablo

doi:10.1007/s11856-012-0163-4

Cited by 9 publications

(34 citation statements)

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“…Therefore, the two A-orbits x A and y A have coprime sizes. By Theorem 1.1 in [4], the A-orbit (xy) A has size |x A | · |y A |. Hence |(xy) G | = |x G | · |y G |, where xy ∈ E is an involution.…”

Section: Proofs Of Theorems a And Bmentioning

confidence: 95%

Real class sizes

Tong-Viet

2018

Isr. J. Math.

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In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group G have the same 2-part and the Sylow 2-subgroup of G satisfies certain condition, then G is solvable.

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Section: Proofs Of Theorems a And Bmentioning

confidence: 95%

Real class sizes

Tong-Viet

2018

Isr. J. Math.

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“…for any irreducible kG-module V with V B = 0 (in cross characteristic, and G is a finite simple group of Lie type as before). For classical groups, any irreducible module V is quasi-equivalent to its dual [DGPS,2.1,2.4] and [TZ] (ii) If Y is an indecomposable summand of M , then the isomorphism class of Y is determined by its socle (or head). (iii) If Y is an indecomposable summand of M , then its socle and head are self-dual kG-modules.…”

Section: This Givesmentioning

confidence: 99%

Sectional rank and cohomology

Guralnick

Tiep

2020

Journal of Algebra

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“…It remains to consider the m = 2 case. The q = 7 case is done in [6]. If q > 7, then the restrictions on q imply that q 17, and then by Lemma 3.5 there exists s ∈ T such that K ∩ K s ∼ = C 2 .…”

Section: When T = Psl(2 Q)mentioning

confidence: 99%

“…Since |T : K 1 | and |T : K 2 | are coprime, it follows from Lemma 3.16 in [11] that T = K 1 K 2 is a maximal coprime factorisation. By the list in [6] it follows that either q is even or q ≡ 3 (mod 4) or q = 29.…”

Section: Recall the Action Of H Onmentioning

confidence: 99%

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Coprime subdegrees of twisted wreath permutation groups

Chua

Giudici

Morgan

2019

Proceedings of the Edinburgh Mathematical Society

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Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the nonabelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q) m , and determine all values of m and q for which G(m, q) has nontrivial coprime subdegrees. In the case where m = 2 and q / ∈ {7, 11, 29} we obtain a full classification of all pairs of nontrivial coprime subdegrees.We can show that N is a subgroup of T H and that N ∼ = T R . Furthermore, the group N is invariant under the action of H, so H acts as a group of automorphisms on N.Definition 2.1. We define the twisted wreath product determined from (T, H, φ) to be the group G = N ⋊ H. The group G acts on Ω = N with N acting by right multiplication and H acting by automorphisms, that is, α nh = (αn) h for all α ∈ Ω, n ∈ N and h ∈ H. Lemma 2.2. The nontrivial subdegrees of G are the values of |H : H f | for f ∈ N\{id}. Also, no nontrivial subdegrees of G are equal to 1.2Proof. We can verify that G id = H, so the nontrivial subdegrees of G are of the form |G id :Since G is not cyclic of prime order, no nontrivial subdegrees of G are equal to 1.The following result from [5, Lemma 4.7A] gives a set of sufficient conditions for a twisted wreath product to be primitive.Theorem 2.3. Let T be a finite nonabelian simple group, and suppose H is a primitive permutation group with point stabiliser L. Suppose that the group of inner automorphisms of T is contained in the image of φ, but Im φ is not a homomorphic image of H. Then the twisted wreath product determined from (T, H, φ) is a primitive group with regular socle N, and N ∼ = T m where m = |H : L|.We will deal with a class of primitive TW groups constructed from a group of diagonal type. Lemma 2.4. Let T be a finite nonabelian simple group, let H = T ≀ S m and let L = {(x, . . . , x)σ | x ∈ T, σ ∈ S m }. Define φ : L → Aut(T ) by setting φ((x, . . . , x)σ) = i x for all x ∈ T and σ ∈ S m , where i x denotes the automorphism of T induced by conjugation by x. Then the construction in Definition 2.1 yields a primitive TW permutation group G(m, T ) with socle isomorphic to T |T | m−1 and point stabiliser isomorphic to T ≀ S m . Proof. The group H acts primitively on the set of right cosets of L. Note that Inn(T ) = Im φ and that Inn(T ) is not a homomorphic image of H. All the conditions of Theorem 2.3 have been satisfied, so G(m, T ) is indeed a primitive group of type TW. Since |H : L| = |T | m ·m! |T |·m! = |T | m−1 , the socle is of the form T |T | m−1 .Remark 2.5. We define G(m, q) = G(m, PSL(2, q)) and note that G (2, 7) is the primitive TW group in [6, p. 12-14] with nontrivial coprime subdegrees.Lemma 2.7. Let D be a maximal subgroup of H. If there exists t ∈ H such that ...

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Coprime subdegrees for primitive permutation groups and completely reducible linear groups

Cited by 9 publications

References 24 publications

Real class sizes

Real class sizes

Sectional rank and cohomology

Coprime subdegrees of twisted wreath permutation groups

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