# Thread: Sizes of kernels of homomorphisms

1. ## 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.

2. ## Re: Sizes of kernels of homomorphisms

Originally Posted by abscissa
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.