Question

If $f:G\to H$ is a homomorphism of finite groups, prove that $\left|Imf\right|$ divides $\left|G\right|$ and $\left|H\right|$. [Im$f$ was defined just before Theorem 7.20.]

Short answer

It is proved that, $\left|\mathrm{Im}f\right|$ divides $\left|G\right|$ and $\left|H\right|$.

Step-by-step solution

First Isomorphism Theorem

Let$f:G\to H$ be a surjective homomorphism of groups with kernel K. Then the quotient group $G/K$ is isomorphic to H.

 |Imf| divides  |G| and  |H|

Let G and H be finite groups and $f:G\to H$ is a homomorphism of finite group.

Since H is a finite group and Im f is a subgroup of H, $\left|\mathrm{Im}f\right|$ divides $\left|H\right|$.

Then,byFirst Isomorphism Theorem, $G/\ker f\cong Imf$.

Since G is finite, $\frac{\left|G\right|}{\left|\ker f\right|}=\left|Imf\right|$ which implies $\left|\ker f\right|=\frac{\left|G\right|}{\left|Imf\right|}$.

Thus,$\left|\mathrm{Im}f\right|$ divides$\left|G\right|$.

Therefore, $\left|\mathrm{Im}f\right|$ divides $\left|G\right|$ and $\left|H\right|$.