Question about order of an element

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I think that I just noticed something that I probably should have noticed a lot earlier in my Abstract Algebra class:

If G is a group which contains some g of finite order, then by definition g^k=e for some integer k. Does this mean that each element of finite order in G can be written as the generator of some some cyclic subgroup of G?
 
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Every element of a group generates a cyclic subgroup of the larger group, and the order of the element doesn't need to be finite for the resulting subgroup to be cyclic. Cyclic just means that it's generated by one element.
 

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