Let be a finite group such that , with prime. Then divides the order of .

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# Group theory is hard

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Let be a finite group such that , with prime. Then divides the order of .

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Let be a Sylow -subgroup of . The order of is for some number greater or equal to . If and have not trivial intersection, we are done. Otherwise, note that (and ). Then . But is abelian, thus for every , the commutator . Thus is abelian and then is abelian. A contradiction.

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