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if p and q are primes, show that every proper subgroup of a group of order...

if p and q are primes, show that every proper subgroup of a group of order pq is cyclic

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Answer #1

Proof:

Let   be a proper subgroup of group .

  by Lagrange's theorem

  

If   then   

    is cyclic.

If      then    is cyclic as a group of prime order is cyclic.

Hence, every proper subgroup of a group of order    is cyclic.

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