|G|=pq then |G| is abelian or Z(G)=1

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In summary, the conversation discusses a proof that states if |G|=pq for primes p and q, then G is either abelian or Z(G)=1. The proof involves showing that |Z(G)| = p or |Z(G)| = q are impossible, and that G/Z(G) must be cyclic of prime order, leading to a contradiction if q>1. This is because groups of prime order are cyclic.
  • #1
ArcanaNoir
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Homework Statement


Show that if [itex]|G|=pq[/itex] for some primes p and q, then G is abelian or Z(G)=1.


Homework Equations



|G| = pq =⇒ |Z(G)| = 1, p, q, or pq. Prove: |Z(G)| = p and |Z(G)| = q are impossible. If
|Z(G)| = p then |G/Z(G)| =|G|/|Z(G)| =pq/p = q. But then, since G/Z(G) is cyclic of prime order,
|G/Z(G)| = 1. Thus q = 1, a contradiction since q > 1. Similarly, |Z(G)| = q =⇒ p = 1, a
contradiction to the definition of p


The Attempt at a Solution



I am trying to understand the part of the proof that says "G/Z(G) is cyclic". Why is this so?
 
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  • #2
ArcanaNoir said:

Homework Statement


Show that if [itex]|G|=pq[/itex] for some primes p and q, then G is abelian or Z(G)=1.


Homework Equations



|G| = pq =⇒ |Z(G)| = 1, p, q, or pq. Prove: |Z(G)| = p and |Z(G)| = q are impossible. If
|Z(G)| = p then |G/Z(G)| =|G|/|Z(G)| =pq/p = q. But then, since G/Z(G) is cyclic of prime order,
|G/Z(G)| = 1. Thus q = 1, a contradiction since q > 1. Similarly, |Z(G)| = q =⇒ p = 1, a
contradiction to the definition of p


The Attempt at a Solution



I am trying to understand the part of the proof that says "G/Z(G) is cyclic". Why is this so?

I believe it's because groups of prime order are cyclic. This stems from Lagrange's theorem.
 
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  • #3
Omg yes. Thank you, I had all those pieces just wasn't putting them together for some reason. :)
 

1. Is every group with order equal to the product of two distinct primes necessarily abelian?

No, not every group with order equal to the product of two distinct primes is necessarily abelian. This is because there exist non-abelian groups with order equal to the product of two distinct primes, such as the symmetric group S3.

2. Can a non-abelian group have an order equal to the product of two distinct primes?

Yes, a non-abelian group can have an order equal to the product of two distinct primes. An example of this is the symmetric group S3, which has an order of 6 (2x3) and is non-abelian.

3. What is the significance of the condition |G|=pq in determining if a group is abelian or not?

The condition |G|=pq, where p and q are distinct primes, is significant because it helps to identify if a group is abelian or not. If a group has an order equal to the product of two distinct primes, then it is either abelian or has a non-trivial center (Z(G)≠1).

4. Can a group with order equal to the product of two distinct primes have a non-trivial center?

Yes, a group with order equal to the product of two distinct primes can have a non-trivial center. This means that the group is not abelian, as the center is a subgroup of the group and an abelian group must have a trivial center (Z(G)=1).

5. What is the relationship between the order of a group and its center in determining if the group is abelian?

The relationship between the order of a group and its center is important in determining if the group is abelian. If the order of a group is equal to the product of two distinct primes, then the group is either abelian or has a non-trivial center. If the order of a group is not equal to the product of two distinct primes, then the group may still be abelian, but this cannot be determined solely based on the order of the group.

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