Permutation representations of finite simple groups: Orbits of cyclic ...
Permutation representations of finite simple groups: Orbits of cyclic ...
Permutation representations of finite simple groups: Orbits of cyclic ...
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Idea <strong>of</strong> the Pro<strong>of</strong><br />
A mixture <strong>of</strong> the two techniques already mentioned is needed:<br />
intersections <strong>of</strong> conjugacy classes <strong>of</strong> sub<strong>groups</strong> and embeddings<br />
<strong>of</strong> permutation modules.<br />
1. Embeddings: Let G be in L1, acting doubly transitively on<br />
the set Ω. Suppose that G also acts non-trivially on some other<br />
set ∆. Consider the corresponding F G-modules F Ω and F ∆.<br />
Let<br />
ϕ: F Ω → F ∆<br />
be a non-zero F G-homomorphism. Then its kernel can only be<br />
0, I or F Ω − I.