I should be able to do this one but I'm stuck...
I have a cyclic subgroup of order where is prime.
Let be a prime dividing . Show G contains a unique cyclic subgroup of order .
Now I know Cauchy's Theorem tells me G indeed has at least one subgroup of order .
Also I know for some positive integer .
I may be making this too complicated...Any hints?
that's not hard to prove:
first of all recall that every subgroup of a cyclic group is cyclic. now let be a group of order and then clearly is a subgroup of of order
now suppose is any subgroup of G of order then and thus i.e. let then and hence
therefore, since we must have