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

**My work:** $\ker(\alpha)

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

$|im\alpha|$ 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)|}
$

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

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.

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

**My work:** $\ker(\alpha)

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

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

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

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

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)

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.