Question

Question: Let $f:G\to H$be a homomorphism of groups and suppose that $a\in G$has finite order K.

(b) Prove that$\left|f\left(a\right)\right|$ divides $\left|a\right|$. [Hint: Exercise 7.9.]

Short answer

It has been proved that $\left|f\left(a\right)\right|$divides $\left|a\right|$.

Step-by-step solution

Step-By-Step SolutionStep 1: Show that fadividesa

Given that$\left|a\right|=k$

Since , $f{\left(a\right)}^k=e$therefore, $f\left(a\right)$order of divides k.

So, order of f(a) divides $\left|a\right|$.

Conclusion

Hence,$\left|f\left(a\right)\right|$ divides $\left|a\right|$.