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Factor groups:part 2

  1. Oct 23, 2007 #1
    1. The problem statement, all variables and given/known data
    PRove that a factor group of an abelian group is abelian

    2. Relevant equations

    3. 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?
  2. jcsd
  3. Oct 23, 2007 #2
    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.
  4. Oct 23, 2007 #3


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    Homework Helper

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

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

    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?

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