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## Main Question or Discussion Point

My textbook (Abstract Algebra by W.E. Deskins) says that the order of an element of a group is equal to the integral power to which it must be raised to equal the identity element. It also says that the order of the generator of a cyclic group is equal to the order of the group it generates.

First of all, can't any single generator generate multiple (if not infinitely many) cyclic groups? It seems it would have more than one order.

Second, does the definition for the order of a generic group element not apply to generators of cyclic groups? It seems there must be some cases where the two definitions will not yield the same value.

There is another portion of the text that says that, for any element of a group, [tex]x^{0}=e[/tex] (e is the identity element)...so the order of every element of every group is zero?

First of all, can't any single generator generate multiple (if not infinitely many) cyclic groups? It seems it would have more than one order.

Second, does the definition for the order of a generic group element not apply to generators of cyclic groups? It seems there must be some cases where the two definitions will not yield the same value.

There is another portion of the text that says that, for any element of a group, [tex]x^{0}=e[/tex] (e is the identity element)...so the order of every element of every group is zero?

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