Class equation and a group of order pq

[simmonj7]

Homework Statement

Use the class equation to show that greup G of order pq, with p and q prime, has an element of order p.

Homework Equations

|G| = $\sum$ |C(x)| (class equation)
Z = center of the group G = {z$\in$G: zy = yz for all y in G}
|G| = |Z(x)||C(x)| (counting formula)

The Attempt at a Solution

So I was told that I should consider this problem in cases. First, assume that G is abelian and then assume G was non abelian.

So when G is abelian, |Z|= pq so Z = G. Since G is finite and any finite abelian group is a product of its Sylow subgroups G = C$_{p}$ x C$_{q}$. So we can then see that G would have an element of order p.

Now, so far I have proved that if G is non abelian, our center has to be trivial i.e. |Z| = 1. Using this, the class equation becomes |G| = 1 + $\sum$ |C(x)| (the sum is for the remaining conjugacy classes). From here is where I get stuck...Since the order of G is pq, the sumation on the right hand side will have to sum to pq -1 (but I have no idea if that is relevant).

Also, I tried assuming that there wasn't an element of order p. Then, every non trivial element x of G would have to be of order q (since it can't be of order 1, pq, or p and those are our only possible orders which divide the order of the group). From there, I can't quite see where to go from here. I know that every term on the right hand side must divide the order of the group, i.e. pq. So, since the order of the remaining conjugacy classes can't be 1 or pq, they have to be either of order p or order q. If the conjugacy class of x was order q, then the centralizer of any element x would have have to be of order p by the counting formula...We know the centralizer would have to contain the center and the element x of order q.

I feel like I am so close but I am just not seeing it at this point. Help please.

 

[trucky]

First, if G is non abelian, |Z|<pq. It is not necessary that |Z|=1.

I think this is correct:
If we assume that G does not have an element of order p, then |Z(x)|=q for all x not in the center, where Z(x) is the centralizer of x. Then |C(x)|=p for all x not in the center. Then the class equation implies that there must be p conjugacy classes of order 1, i.e., |Z|=p, a contradiction. So G has an element of order p.

 

[simmonj7]

What you said is false.
If G is a non abelian group of order pq where p and q are distinct primes, the center is trivial. I proved this is homework and I know this for a fact.

Consider the quotient group G/Z. Suppose that Z is a non-trivial subgroup then its order is either p or q (because it can't be pq because then it would be abelian and can't be 1 because then it wouldn't be non trivial). So, the order of G/Z is either q or p. We know that every group of prime order is cyclic, so G/Z must be cyclic. But if G/Z is cyclic, then G is Abelian which is a contradiction.

There's your proof for why the center must be trivial.

 

[trucky]

My bad, you're totally correct. The reasoning for existence of an element of order p still goes through though. |Z|=p is still a contradiction.

 

[Dick]

I think you already basically have it, don't you? If you assume there is no element of order p then all of the conjugacy classes have to have p elements, because as you said, a conjugacy class with q elements would give a centralizer subgroup with p elements. So now what does the class equation tell you?

 

[simmonj7]

How does there not being an element of order p imply that the centralizer is of order q? If every non identity element isn't of order p, it obviously has to be of order q since it can't be of order 1 or pq. Then the centralizer of an element x would have to be at least size q (<x> must be contained in Z(x) ) but it can not be all of pq. Are you assuming that p<q so that you can say that because then <x> would be maximal?

Also, I don't see how you are saying that the conjugacy classes being of order p implies that there must be p conjugacy classes of order 1? If the conjugacy class of every other element is of order p, doesn't the class equation imply that pq = 1 + np where n is some integer?

 

[Dick]

How does there not being an element of order p imply that the centralizer is of order q? If every non identity element isn't of order p, it obviously has to be of order q since it can't be of order 1 or pq. Then the centralizer of an element x would have to be at least size q (<x> must be contained in Z(x) ) but it can not be all of pq. Are you assuming that p<q so that you can say that because then <x> would be maximal?

Also, I don't see how you are saying that the conjugacy classes being of order p implies that there must be p conjugacy classes of order 1? If the conjugacy class of every other element is of order p, doesn't the class equation imply that pq = 1 + np where n is some integer?

I thought you had spelled this out in your original post. But if x is an element with a conjugacy class containing q elements, then doesn't the centralizer subgroup of x contain p elements? Contradicting the statement there is no element of order p? Maybe I was reading between the lines? And you are claiming I said some stuff I didn't say. But I do agree that finally the class equation would say pq=1+np. That doesn't work.

 

[simmonj7]

Dick,
I wasn't responding to your post. I get what you are saying and yes I had spelled out what you said in my original post. I have the problem solved now. I just didn't understand the logic of the other person who was posting.

Thanks.

 

[Dick]

Dick,
I wasn't responding to your post. I get what you are saying and yes I had spelled out what you said in my original post. I have the problem solved now. I just didn't understand the logic of the other person who was posting.

Thanks.

Oh, ok. Sometimes good to quote the post you are responding to. Very welcome.