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.) 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. THANK YOU
Proof: Let G 1 , .., G t be a system of representative for the conjugacy classes of subgroups of G. Let Ω i := G/G i be the corresponding G-sets, obtained by right multiplication. Suppose Ω = ∑ n i Ω i and Ω ′ = ∑ n ′ i Ω i are G-sets with fix Ω (U) = fix Ω ′(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 and 38: Comment 2: Moving on from cyclic su
Page 39: Here the lemma for permutation modu
Page 43: Proof: Let G 1 , .., G t be a syste

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

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