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Math Help - Nilpotent Groups w/ Normal Subgroups

  1. #1
    Senior Member roninpro's Avatar
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    Nilpotent Groups w/ Normal Subgroups

    Hello. I have been asked to prove the following:

    If N is a nontrivial normal subgroup of a nilpotent group G, then N \cap Z(G) is nontrivial.

    I attempted to take the factor group G/N (since it too is nilpotent) and try to deal with the image of Z(G) under the canonical homomorphism, but I can't get the result to come out.

    This is extremely frustrating, and thus, I would appreciate any suggestions you may have. (Though, I am not necessarily searching for a complete solution.)

    Thank you.
    Last edited by roninpro; November 28th 2009 at 10:11 PM. Reason: Incorrect tags for LaTeX formatting. Corrected the statement of the problem.
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  2. #2
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    Quote Originally Posted by roninpro View Post
    Hello. I have been asked to prove the following:

    If N is a normal subgroup of a nilpotent group G, then N \cap Z(G) is nontrivial.

    I attempted to take the factor group G/N (since it too is solvable) and try to deal with the image of Z(G) under the canonical homomorphism, but I can't get the result to come out.

    This is extremely frustrating, and thus, I would appreciate any suggestions you may have. (Though, I am not necessarily searching for a complete solution.)

    Thank you.
    you forgot to mention the condition N \neq \{1 \}. let \{1 \}=G_0 \subset G_1 \subset \cdots \subset G_n=G be the upper central series of G. since N \cap G_n=N \neq \{ 1 \}, we can define k=\min \{j : \ N \cap G_j \neq \{ 1 \} \}.

    note that k \geq 1 because N \cap G_0=\{ 1 \}. so N \cap G_k \neq \{1\} and N \cap G_{k-1} = \{ 1 \}. choose 1 \neq g \in N \cap G_k. then, since gG_{k-1} \in G_k/G_{k-1} = Z(G/G_{k-1}), we have gxG_{k-1}=xgG_{k-1}, for all

    x \in G. therefore y=g^{-1}x^{-1}gx \in G_{k-1}. but, since N is normal and g \in N, we also have y \in N. thus y \in N \cap G_{k-1}=\{1\}. that means gx=xg, for all x \in G. hence g \in N \cap Z(G). \ \Box
    Last edited by NonCommAlg; November 28th 2009 at 09:30 PM.
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  3. #3
    Senior Member roninpro's Avatar
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    Sorry for the mistakes - I typed up the post in a hurry.

    I originally went about halfway in the direction of your solution but couldn't see what to do with it, so I discarded it, unfortunately.

    I see how to do it now. Thank you very much for your response.
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