I have to prove that every group of order 30 has an order of 15, without using Sylow's theorems. My attempt is to use the Cauchy theorem and from there I obtain that there are elements with order 2, 3 and 5. But I don't know how to conclude that there is an element of order 15.
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It is enough to prove that there is a subgroup of order $15$. See https://math.stackexchange.com/questions/208760/simple-method-the-show-that-a-group-of-order-15-is-cyclic. – lhf Apr 09 '18 at 14:20
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Thanks for the help! – Gabriela Torres Apr 09 '18 at 15:43