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Math Help - 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 \alpha:G \rightarrow H, \beta:H \rightarrow K be group homomorphisms. Which is larger, \ker(\beta\alpha)$ or $\ker(\alpha)?**


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

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


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


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


    |ker(\beta\alpha)|=\frac{|G|}{|im(\beta\alpha)|}, |ker(\alpha)|=\frac{|G|}{|im(\alpha)|}<br />

    \frac{|\ker(\beta\alpha)|}{|\ker\alpha|} \leq \frac{|H|}{|im(\beta\alpha)|}<br />

    With similar analysis, I get |\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; November 29th 2013 at 06:49 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 \alpha:G \rightarrow H, \beta:H \rightarrow K be group homomorphisms. Which is larger, \ker(\beta\alpha)$ or $\ker(\alpha)?**


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

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


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


    |ker(\beta\alpha)|=\frac{|G|}{|im(\beta\alpha)|}, |ker(\alpha)|=\frac{|G|}{|im(\alpha)|}<br />

    \frac{|\ker(\beta\alpha)|}{|\ker\alpha|} \leq \frac{|H|}{|im(\beta\alpha)|}<br />

    With similar analysis, I get |\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:** \ker(\alpha)<G, \ker(\beta\alpha)<G, |G|=|\ker\alpha|*|im\alpha|=|\ker\beta|*|im\beta|

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

    So the rest of your progress is no progress.

    This is quite simple.

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