# Thread: Abelian group, number of elements

1. ## Abelian group, number of elements

Hello, I am stuck with the following question and was wondering if anyone could give me some advice.

Let be an abelian group of order pq, where p<q are primes. Show that there are exactly p-1 elements of order q, q-1 elements of order q, and (p-1)(q-1) elements of order pq.

OK so we have a theorem which tells us that there must be only one subgroup of order q (which is normal). This subgroup is cyclic because it is of prime order, so every non identity element is of order q. This gives q-1 elements of order q. There can be no other elements of order q in G since they would generate a distinct subgroup of order q, and there can be only one.

So there are definitely only q-1 elements of order q in G.

By Cauchy's theorem, there exists an element of order p in G. This generates a cyclic group of order p which has p-1 elements of order p. why can there be no more than that?
Knowing this part implies the number of elements of order pq = pq - (p-1) - (q-1) = (p-1)(q-1), so how do I show that there are no more than p-1 elements of order p?

Any help with this would be appreciated, Thank you

2. Originally Posted by slevvio
Hello, I am stuck with the following question and was wondering if anyone could give me some advice.

Let be an abelian group of order pq, where p<q are primes. Show that there are exactly p-1 elements of order q, q-1 elements of order q, and (p-1)(q-1) elements of order pq.

OK so we have a theorem which tells us that there must be only one subgroup of order q (which is normal). This subgroup is cyclic because it is of prime order, so every non identity element is of order q. This gives q-1 elements of order q. There can be no other elements of order q in G since they would generate a distinct subgroup of order q, and there can be only one.

So there are definitely only q-1 elements of order q in G.

By Cauchy's theorem, there exists an element of order p in G. This generates a cyclic group of order p which has p-1 elements of order p. why can there be no more than that?
Knowing this part implies the number of elements of order pq = pq - (p-1) - (q-1) = (p-1)(q-1), so how do I show that there are no more than p-1 elements of order p?

Any help with this would be appreciated, Thank you
Hint: an abelian group of order $\displaystyle pq,$ with $\displaystyle p \neq q$ primes, is cyclic.

3. I see why this would imply the result

but why is the group cyclic ?

4. Originally Posted by slevvio
I see why this would imply the result

but why is the group cyclic ?

Because

1) It is the direct product of its Sylow subgroups

2) If $\displaystyle C_r$ denotes the cyclic group of order $\displaystyle r\in\mathbb{N}$ , then $\displaystyle lcm(m,n)=1\Longrightarrow C_m\times C_n\cong C_{mn}$

Tonio