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

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Therefore, given g ∈ Sym n with regular orbits, can we identify the primitive almost simple groups G with 〈 g 〉 ⊂ G ⊂ Sym n The problem I first mentioned is a key step towards answering such questions. I will report about joint work with Alex Zalesskii. Initial Comments [1] : Let C, B ⊆ A be finite groups. Then C has a regular orbit on the cosets of B in A if and only if C ∩ B a ⊆ K for some a ∈ A, where K is the core of B in A.
For cyclic C let the square-free part of the action be the subgroup C ′ such that if the prime p divides |C : C ∩ K| then p divides |C ′ : C ′ ∩ K| but not p 2 . A simple reduction is Lemma: C has a regular orbit on the cosets of B in A if and only if C ′ has a regular orbit on this set. There is much work on intersections of conjugacy classes of subgroups, and in some sense this is the main task in tackling the problem.
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: Therefore, given g ∈ Sym n with r
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 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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