Question

Question: If G is an abelian group, prove that the function $f:G\to G$given by $f\left(x\right)=x^2$ is a homomorphism.

Short answer

It has been proved that $f\left(x\right)=x^2$is a homomorphism

Step-by-step solution

Step-By-Step SolutionStep 1: Prove that fab=fafb

Consider $f:G\to G$given by $f\left(x\right)=x^2$.

Now,

$$\begin{gathered} f\left(ab\right)={\left(ab\right)}^2 \\ =abab \end{gathered}$$

Since G is abelian$ba=ab$,

So,

Thus,

$$\begin{gathered} f\left(ab\right)=aabb \\ =a^2{b}^2 \\ =f\left(a\right)f\left(b\right) \end{gathered}$$

Conclusion

Since, $f\left(ab\right)=f\left(a\right)f\left(b\right)$Thus, fis a homomorphism.