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

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[2] : Let (G, Ω) be an action of the group G on the set Ω. Denote F := C and consider the permutation module F Ω for F G. For cyclic groups the permutation module is crucial in detecting regular orbits.
[2] : Let (G, Ω) be an action of the group G on the set Ω. Denote F := C and consider the permutation module F Ω for F G. For cyclic groups the permutation module is crucial in detecting regular orbits. Lemma (Brauer, 1940): Two n × n permutation matrices are conjugate in GL(n, C) if and only if the corresponding permutations are similar (the same up to renaming) in Sym n . In other words: for cyclic C we have F Ω ∼ = F ∆ as F C -modules if and only if (C, Ω) ∼ = (C, ∆) as G-sets. This lemma turns out to be very useful for our purposes.
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: For cyclic C let the square-free pa
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 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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