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

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Here the lemma for permutation modules of cyclic groups can not be applied any longer. Instead we need to recall a theorem of Burnside that is not so well-known any longer. If G acts on Ω and if U ⊆ G let fix Ω (U) be the number of elements ω ∈ Ω fixed by all elements of U. (This function extends the permutation character to subgroups.)
Here the lemma for permutation modules of cyclic groups can not be applied any longer. Instead we need to recall a theorem of Burnside that is not so well-known any longer. If G acts on Ω and if U ⊆ G let fix Ω (U) be the number of elements ω ∈ Ω fixed by all elements of U. (This function extends the permutation character to subgroups.) Theorem (Burnside, 1911): Let (G, Ω) and (G, ∆) be permutation actions. Then (G, Ω) = ∼ (G, ∆) if and only if fix Ω (U) = fix ∆ (U) for all subgroups U in G.
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 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: Comment 2: Moving on from cyclic su
Page 41 and 42: Proof: Let G 1 , .., G t be a syste
Page 43: Proof: Let G 1 , .., G t be a syste

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