Proving G/H is Abelian: Normal Subgroups in Abelian Groups

In summary, the student was asked to prove that if H is a normal subgroup of an abelian group G, then G/H is abelian. Their proof began by stating that a and b are elements of G/H, but the instructor expects to find errors in the rest of the proof as a and b should be represented as cosets of H in G. Some authors use the notation [g] = Hg to represent elements that are "H equivalent" to g. The student then realizes their mistake and agrees that it makes more sense to forget about a and b.
  • #1
ehrenfest
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[SOLVED] group theory question

Homework Statement


A student is asked to show that if H is a normal subgroup of an abelian group G, then G/H is abelian. THe student's proof starts as follows:

"We must show that G/H is abelian. Let a and b be two elements of G/H."

Why does the instructor reading this proof expect to find nonsense from here on in the student's paper?

Homework Equations


The Attempt at a Solution


That's probably how I would start my proof...I don't see anything wrong with it.
 
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  • #2
Elements of G/H look like Hg, i.e. they are cosets of H in G. Some authors use the notation [g] = Hg to stand for the elements that are "H equivalent" to g.
 
  • #3
Yeah. I would have then said a must be equal to xH and b must be equal to yH, but I guess it really makes more sense to forget about a and b. Thanks.
 

1. Why is it important to prove that G/H is abelian?

Proving that G/H is abelian is important because it helps us understand the structure of the group G/H and its subgroups. It also allows us to make certain conclusions about the properties and behavior of the elements in G/H, which can be useful in solving various problems.

2. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is closed under conjugation by elements of the group. In other words, if H is a subgroup of G, and g is an element of G, then gHg^-1 is also a subgroup of G. This property is important in proving that G/H is abelian.

3. How do you prove that G/H is abelian?

To prove that G/H is abelian, we need to show that the cosets of H in G commute with each other. This can be done by showing that for any two cosets aH and bH, their product (aH)(bH) is equal to (bH)(aH). This can be simplified to show that abH = baH. If this is true for all elements a and b in G, then G/H is abelian.

4. What are some examples of abelian groups?

Some examples of abelian groups include the integers under addition, the real numbers under addition, and the group of 2x2 matrices with real entries and determinant 1 under matrix multiplication. Any finite cyclic group is also abelian.

5. Can a group be both abelian and non-abelian?

No, a group cannot be both abelian and non-abelian. A group is either abelian, meaning all of its elements commute with each other, or it is non-abelian, meaning there exist elements that do not commute. A group cannot have both of these properties simultaneously.

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