Question about generator of cyclic group

[dumbQuestion]

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^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 a⁵=e, but a¹⁰=m?


Thanks

 

[Number Nine]

What happens with a^s+1? Well, you get a^sa = ea = a, so you start the same cycle over again.

 

[dumbQuestion]

Number Nine, thank you very much for this, that makes complete sense and clears my confusion and I don't know why I didn't think of it this way before!