Question

Question: If $f:G\to H$is an injective homomorphism of groups and $a\in G$, prove that $\left|f\left(a\right)\right|=\left|a\right|$

Short answer

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

Step-by-step solution

Step-By-Step SolutionStep 1: Show that G≅Imf

Given that$f:G\to H$ is an injective homomorphism of groups.

Using Theorem 7.20, we get$G\cong Imf$.

Show that fa=a

Now,$G\cong Imf$

And $a\in G$, therefore,$\left|f\left(a\right)\right|=\left|a\right|$

Conclusion

Hence ,$\left|f\left(a\right)\right|=\left|a\right|$