Say we have a cyclic group G, and a generator a in G. This means [a] = G. We know the order of an element a, is the order of the group it generates, [a], and also this is the smallest integer s such that a(adsbygoogle = window.adsbygoogle || []).push({}); ^{s}=e, where e is the identity element. In this case, [a]=G, so s is just the order of G.

Now my question is, since a is a generator of G, this means there is an integer t such that for every m in G, a_{t}=m. But is it always true that t ≤ s? What I mean is, does the integer s such that a^{s}= e (the identity) always have to be greater than the integer t such that a^{t}=m (where m is just any old element in G, NOT the identity!) Could you have a situation say where you have a group G and generator a, and say e is the identity element in G, and m is some other element in G, and a^{5}=e, but a^{10}=m?

Thanks

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# Question about generator of cyclic group

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