# Orders of Cyclic Groups, Prime Order and Subgroups

(1) Let G be cyclic of order pn, p prime. Let H,K < G. Show that either H C K or K C H.

(2) Let G 6 ≠ {e} such that it has no proper subgroups. Then G must be cyclic of prime order.

(3) If G is a group with order pq where p > q are primes and q does not divide p − 1, then G must be cyclic.

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#### Solution Summary

Orders of cyclic groups, prime order and subgroups are investigated. The solution is detailed and well presented.

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