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

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Embeddings Comment 1: As an example, let G = P SL(n, p) acts on Ω = points of projective space. Let ∆ be any other G-set, and assume that we are in the embedding case. Take some g ∈ G. How does g act on ∆ For instance, if g is a transvection then some part of ∆ looks exactly like Ω : many fixed points, with the remainder all cycles of length p. Similarly, let g be a Singer cycle. Again, some part of ∆ looks exactly like Ω : at least one cycle of length (p n − 1)/(n − 1), with other shorter cycles, etc.
Embeddings Comment 1: As an example, let G = P SL(n, p) acts on Ω = points of projective space. Let ∆ be any other G-set, and assume that we are in the embedding case. Take some g ∈ G. How does g act on ∆ For instance, if g is a transvection then some part of ∆ looks exactly like Ω : many fixed points, with the remainder all cycles of length p. Similarly, let g be a Singer cycle. Again, some part of ∆ looks exactly like Ω : at least one cycle of length (p n − 1)/(n − 1), with other shorter cycles, etc. QUESTION 1: Is is possible to identify G from such orbit statistics
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 and 24: Idea of the Proof A mixture of the
Page 25 and 26: Proposition: Let G be doubly transi
Page 27 and 28: Proposition: Let G be doubly transi
Page 29 and 30: 2. Intersections of conjugacy class
Page 31: 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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