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It is rather obvious that every Hall π-subgroup is a Sylow π-subgroup. In general, however, G need not contain any Hall π-subgroups. For example, a Hall {3,5}-subgroup of $A_5$ would have index 4, but $A_5$ have no such subgroups. (Why?)

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    Are you asking how you know what $A_5$ has no subgroups of index $4$?2012-03-06

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