Proving Abelian Factor Groups: Group Theory Homework Solution

[Benzoate]

Homework Statement

PRove that a factor group of an abelian group is abelian

Homework Equations

The Attempt at a Solution

Assume G in G/H is abelian. Let there be elements a and b that are in G. such that ab=ba. Since H is a subgroup of G , elements a and b are also in H. Then (aH)(bH)=ab(H) =ba(H)=(bH)(aH) . Therefore , factor group G/H is abelian.

Is my proof correct?

 

[ZioX]

G in G/H is abelian? How about 'G is abelian'. Let there be elements in a and b that are in G such that ab=ba? How about 'Since G is abelian any arbitrary choice of a and b in G will commute'. Since H is a subgroup of G, elements a and b are also in H? There is no injection between G and H unless G=H (H is a subset of G). What you want to say is that every element a in G corresponds to an a+H in G/H.

You have the right idea though.

 

[CompuChip]

Assume G in G/H is abelian.

What do you mean? G is abelian (given), G/H is abelian (trying to prove that). G is not in G/H

Let there be elements a and b that are in G. such that ab=ba.

ab = ba is automatically true if a and b are in G, yes.

Since H is a subgroup of G , elements a and b are also in H.

If and only if G = H of course, then G/H = {1} which is abelian. So the statement is false, but do you need it?

Then (aH)(bH)=ab(H) =ba(H)=(bH)(aH) . Therefore , factor group G/H is abelian.

This looks like the key line in the proof. It's ok, just check what assumptions you need, reviewing the lines above.