Yes. That is in fact what I was refering to. The fundamental theorem of cylic groups. So basically my proof was based on two things

1.

is prime, therefore by LT any element

of

must have order

.

2. Using that fact we know that there are three possiblities for any

. Namely:

or

,

where

. If the first case is true then we are done. If the second is true then

is a cyclic subgroup of

of order

and using the fundamental theorem of cyclic groups we can conclude there is a subgroup of

of order

, which of course means that the gnerator of that group is of order

. If the third case is true then

itself is cyclic, in which case we may apply the fundamental theorem again.

Is there something wrong with that argument?