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Let p,q be prime numbers, not necessarily distinct. If a group G has order pq, prove...

Let p,q be prime numbers, not necessarily distinct. If a group G has order pq, prove that any proper subgroup (meaning a subgroup not equal to G itself) must be cyclic. Hint: what are the possible sizes of the subgroups?

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Answer #1

|G| = pq, where p & q are primes

If H is a Subgroup of G then, by Lagrange's theorem, |H| must divide |G| = pq

So, if, H is proper (and non-trivial) then, |H| = p or q, i.e. H Zp or H Zq

So, in both the cases, H is cyclic.

Even if H is trivial, i.e. H = {e} then also H is cyclic by default.

If, p = q, then, |G| = p² and |H| = p, so, H Zp

So, any proper Subgroup of G is cyclic.

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