Are All Groups of Order pq Nilpotent When p and q Are Primes?

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Groups of order pq, where p and q are distinct primes, are nilpotent if and only if p divides q-1. This condition ensures that the Sylow subgroups can be combined to form the entire group. The discussion emphasizes the importance of understanding the structure of these groups in relation to their Sylow subgroups. Additionally, it highlights that the characterization of primes p and q is crucial for determining nilpotency. Overall, the relationship between group order and nilpotency is a key focus in this mathematical inquiry.
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characterize primes p and q for which each group of order pq is nilpotent
 
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charlamov said:
characterize primes p and q for which each group of order pq is nilpotent


Hint: A finite group is nilpotent iff it is the direct product of its Sylow subgroups, so this leaves you almost without choice...

DonAntonio
 

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