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If $G$ is a group of square-free order with at-least three prime factors, $|G|=p_1p_2....p_r$, $(p_i (2< p_i < p_{i+1})$, does $G$ contain a cyclic subgroup of composite order?

(As groups of square-free order are solvable, $G$ will necessarily have a proper subgroup of composite order.)

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# Subgroups of groups of Square-free order

If $G$ is a group of square-free order with at-least three prime factors, $|G|=p_1p_2....p_r$, $(p_i < p_{i+1})$, does $G$ contain a cyclic subgroup of composite order?

(As groups of square-free order are solvable, $G$ will necessarily have a proper subgroup of composite order.)