Thread: cyclic subgroup

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?

Originally Posted by ziggychick

I should be able to do this one but I'm stuck...

I have a cyclic subgroup of order where is prime. i guess you meant "group"!

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?

something is not right here! any finite cyclic group of order has a unique (cyclic) subgroup of any order which divides so, in your problem, we don't need to assume that p, q are prime .

are you sure G is cyclic?

Originally Posted by NonCommAlg

any finite cyclic group of order has a unique (cyclic) subgroup of any order which divides

Ok, i need to prove the above statement.

I was assuming p and q where prime because this is actually part of a larger proof. ( that there are p distinct homomorphisms from to where p divides q-1.)

I just got stuck on proving has a unique subgroup of order .

Originally Posted by ziggychick

Ok, i need to prove the above statement.

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

ok got it. thanks