Permutation representations of finite simple groups: Orbits of cyclic ...

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Idea of the Proof A mixture of the two techniques already mentioned is needed: intersections of conjugacy classes of subgroups and embeddings of permutation modules. 1. Embeddings: Let G be in L1, acting doubly transitively on the set Ω. Suppose that G also acts non-trivially on some other set ∆. Consider the corresponding F G-modules F Ω and F ∆. Let ϕ: F Ω → F ∆ be a non-zero F G-homomorphism. Then its kernel can only be 0, I or F Ω − I.
Proposition: Let G be doubly transitive on Ω and let ∆ be a transitive G-set. Then exactly one of the following is true: (i) There exists an injective F G-homorphism ϕ: F Ω → F ∆, (ii) G = G ω · G δ for all ω ∈ Ω and δ ∈ ∆.
Page 1 and 2: Permutation representations of fini
Page 3 and 4: Let n ≥ k be integers, k not a pr
Page 5 and 6: The answer is that there should be
Page 7 and 8: Therefore, given g ∈ Sym n with r
Page 9 and 10: For cyclic C let the square-free pa
Page 11 and 12: [2] : Let (G, Ω) be an action of t
Page 13 and 14: It is a special case of a Theorem o
Page 15 and 16: The key observation is therefore th
Page 17 and 18: Main Theorem Let G be an almost sim
Page 19 and 20: L2: G is a classical group, not alr
Page 21 and 22: It is expected that the theorem hol
Page 23: Idea of the Proof A mixture of the
Page 27 and 28: Proposition: Let G be doubly transi
Page 29 and 30: 2. Intersections of conjugacy class
Page 31 and 32: Embeddings Comment 1: As an example
Page 33 and 34: Embeddings Comment 1: As an example
Page 35 and 36: Comment 2: Moving on from cyclic su
Page 37 and 38: Comment 2: Moving on from cyclic su
Page 39 and 40: Here the lemma for permutation modu
Page 41 and 42: Proof: Let G 1 , .., G t be a syste
Page 43: Proof: Let G 1 , .., G t be a syste

Permutation representations of finite simple groups: Orbits of cyclic ...

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