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So we need to prove that the socle of a primitive permutation group is a direct product of isomorphic simple groups.

Now socle means product of the minimal normal subgroups. I know that every non-trivial normal subgroup of a primitive group is regular. But why are two minimal normal subgroups isomorphic?

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If there is more than one minimal normal subgroup, then they centralize each other, but they are both transitive, so that implies that they are both regular and isomorphic to each other. (So there can be at most two minimal normal subgroups.)

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