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May 17th, 2019, 04:47 AM   #1
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Quotient group

If G/H is abelian, where H is a normal subgroup of G, then H is abelian. True or False.

This question seems confusing, as I know if G/H is abelian, then it cannot be deduced whether G is abelian or not. That is it may happen that, if G/H is abelian, then G may not be so. But how to check whether H will be abelian or not. I know that H is a normal subgroup and all normal subgroups are not commutative. H is simply a normal subgroup, therefore nothing can be deduced for H. The statement should stand as false.

Is my attempt correct? or Incorrect?

Another perspective:
If G/H is abelian, then G may or may not be abelian. Suppose, if G is abelian, then every subgroup of an abelian group is abelian and therefore H will
be abelian. However, if G is non-abelian, then I can't say anything about the nature of H and hence the statement is incorrect.

Is my attempt correct? or Incorrect?

Last edited by shashank dwivedi; May 17th, 2019 at 04:51 AM.
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May 17th, 2019, 06:56 PM   #2
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To prove the statement false, all you need is one example. So let $G=S_4$ and $H=A_4$.
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