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Math Help - Question in group theory.

  1. #1
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    Question in group theory.

    Hi all,
    A student I'm helping gave me a question.

    Q: Give an example of a group G with a normal subgroup N such that N and G/N are abelian, but G is not abelian.

    I'm fairly sure there is a dihedral group with this property but i don't have time to find an explicit example.

    Can someone help please?

    Side note: If we weaken the conditions slightly and replace abelian by solvable no example exists.
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  2. #2
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    Quote Originally Posted by whipflip15 View Post
    Hi all,
    A student I'm helping gave me a question.

    Q: Give an example of a group G with a normal subgroup N such that N and G/N are abelian, but G is not abelian.

    I'm fairly sure there is a dihedral group with this property but i don't have time to find an explicit example.

    Can someone help please?

    Side note: If we weaken the conditions slightly and replace abelian by solvable no example exists.
    every dihedral group D_n, \ n \geq 3, satisfies the condition. because D_n=<a,b: \ a^2=b^n=1, \ ab=b^{-1}a>, and so you just choose N=<b>.
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