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Thread: Sizes of kernels of homomorphisms

  1. #1
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    Sizes of kernels of homomorphisms

    I have a problem that I have been stuck on for two hours. I would like to check if I have made any progress or I am just going in circles.


    **Problem: Let $\displaystyle \alpha:G \rightarrow H, \beta:H \rightarrow K$ be group homomorphisms. Which is larger, $\displaystyle \ker(\beta\alpha)$ or $\ker(\alpha)$?**


    **My work:** $\displaystyle \ker(\alpha)<G, \ker(\beta\alpha)<G, |G|=|\ker\alpha|*|im\alpha|=|\ker\beta|*|im\beta|$

    $\displaystyle |G|=|\ker\alpha|[G:\ker\alpha]=|\ker\beta|[G:\ker\beta]$


    $\displaystyle |im\alpha|$ divides $|G|$ and $|H|$


    $\displaystyle |im\beta\alpha|$ divides $|G|$ and $|K|$


    $\displaystyle |ker(\beta\alpha)|=\frac{|G|}{|im(\beta\alpha)|}, |ker(\alpha)|=\frac{|G|}{|im(\alpha)|}
    $

    $\displaystyle \frac{|\ker(\beta\alpha)|}{|\ker\alpha|} \leq \frac{|H|}{|im(\beta\alpha)|}
    $

    With similar analysis, I get $\displaystyle |\ker(\beta\alpha)| \geq \frac{|G|}{|K|}$ and $|\ker(\alpha)| \geq \frac{|G|}{|H|}$.


    This seems like too much work with zero output.
    Last edited by abscissa; Nov 29th 2013 at 05:49 PM.
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  2. #2
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    Re: Sizes of kernels of homomorphisms

    Quote Originally Posted by abscissa View Post
    I have a problem that I have been stuck on for two hours. I would like to check if I have made any progress or I am just going in circles.


    **Problem: Let $\displaystyle \alpha:G \rightarrow H, \beta:H \rightarrow K$ be group homomorphisms. Which is larger, $\displaystyle \ker(\beta\alpha)$ or $\ker(\alpha)$?**


    **My work:** $\displaystyle \ker(\alpha)<G, \ker(\beta\alpha)<G, |G|=|\ker\alpha|*|im\alpha|=|\ker\beta|*|im\beta|$

    $\displaystyle |G|=|\ker\alpha|[G:\ker\alpha]=|\ker\beta|[G:\ker\beta]$


    $\displaystyle |im\beta\alpha|$ divides $|G|$ and $|K|$


    $\displaystyle |ker(\beta\alpha)|=\frac{|G|}{|im(\beta\alpha)|}, |ker(\alpha)|=\frac{|G|}{|im(\alpha)|}
    $

    $\displaystyle \frac{|\ker(\beta\alpha)|}{|\ker\alpha|} \leq \frac{|H|}{|im(\beta\alpha)|}
    $

    With similar analysis, I get $\displaystyle |\ker(\beta\alpha)| \geq \frac{|G|}{|K|}$ and $|\ker(\alpha)| \geq \frac{|G|}{|H|}$.


    This seems like too much work with zero output.
    **My work:** $\displaystyle \ker(\alpha)<G, \ker(\beta\alpha)<G, |G|=|\ker\alpha|*|im\alpha|=|\ker\beta|*|im\beta|$

    How do you get the last equality? $\displaystyle |\ker\beta|\cdot|im\beta|=|H|$, not $\displaystyle |G|$

    So the rest of your progress is no progress.

    This is quite simple.

    $\displaystyle \ker(\alpha)\subseteq\ker(\beta\alpha)$ left to you.
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