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Since 15 divides the order of the cyclic group G, there is exactly one
subgroup H of order 15. (For example H could be Z15.) Each
of the elements in H has an order which divides 15, so each element satisfies X^15 = e. So
there are 15 elements in G such that X^15 = e.
Similarly there are 20 solutions for X^20 = e.
In general, when n divides the order of a cyclic group, the
number of solutions of X^n = e is n.
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Oct 19th, 2015
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