Proving Abelian p-Group: gnhn = (gh)n

  • Thread starter *FaerieLight*
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In summary, the conversation discusses proving the equation gnhn = (gh)n for a p-group G where all elements have order p. The goal is to prove that G is Abelian without using centres. The attempt involves supposing gnhn = (gh)m and trying to prove that m must be congruent to n modulo p. The conversation suggests thinking about generators and using the fact that elements in G have order p. However, a counterexample is given to show that the equation is not always true.
  • #1
*FaerieLight*
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Homework Statement



How would I prove that gnhn = (gh)n, for g,h [itex]\in[/itex]G where G is a p-group, and all its elements have order p?

The Attempt at a Solution



My aim is to prove this in order to prove that G is Abelian, but I don't want to prove it using centres. I've supposed that gnhn = (gh)m for some m, and now I'm trying to prove that m must be congruent to n modulo p, as this is the only way that (gh)m = (gh)n since gh has order p. And this is where I strike a wall, so to speak. Is there a better way to prove that G is Abelian, if this isn't good enough?

Thanks!
 
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  • #2
Think: generator.
 
  • #3
As in...gh generates a cyclic group?
 
  • #4
Well, yes, but don't think about gh. You know that if g is in G, then g has order p, correct?
 
  • #5
That is, can you think of a way to write any element of G in terms of g?
 
  • #6
Yeah, what you're trying to prove is false. A counterexample is given by

[itex](\mathbb{Z}_3\rtimes \mathbb{Z}_3)\times \mathbb{Z}_3[/itex]

All (non-identity) elements have order 3 and the group is non-abelian.
 

1. What is an Abelian p-group?

An Abelian p-group is a mathematical structure that satisfies the properties of being a group (closed under multiplication, contains an identity element, and every element has an inverse) and also has the additional property of being Abelian, meaning that the order of multiplication does not affect the result. P-groups are also defined by being of prime order, meaning that the number of elements in the group is a prime number.

2. What does "gnhn = (gh)n" mean?

This expression is known as the commutator identity and it is a way to show that a group is Abelian. It means that for any elements g and h in the group, the result of multiplying them and then raising the product to the power of n is the same as raising each individual element to the power of n and then multiplying them together.

3. How do you prove that a group is Abelian using "gnhn = (gh)n"?

To prove that a group is Abelian using the commutator identity, you must first show that the identity holds for any two elements in the group. This can be done by simplifying the expression and showing that both sides are equal. Then, you must show that the identity also holds for any power n. If both of these conditions are met, then the group is proven to be Abelian.

4. Can "gnhn = (gh)n" be used to prove that any group is Abelian?

No, the commutator identity can only be used to prove that a group is Abelian if the group is a p-group. It is not a general method for proving the Abelian property of any group.

5. Are there other ways to prove that a group is Abelian?

Yes, there are other methods for proving that a group is Abelian. One way is to show that all of the elements in the group commute, meaning that the order of multiplication does not affect the result. Another way is to show that the group is isomorphic to a known Abelian group, meaning that it has the same structure and properties. The commutator identity is just one specific method for proving the Abelian property of a p-group.

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