Let $G$ be a group show that $φ:G\rightarrow G$ is a group homomorphism

Question

The function $φ:G\rightarrow G$ defined by $φ(a) = a^2$ is a group homomorphism.

I know the definition of a group homomorphism but I don't know how to manipulate the given information to complete the proof.

Answer 1

The statement holds if the group is abelian, because then\begin{align}\phi(ab)&=(ab)^2\\&=(ab)(ab)\\&=a\bigl(b(ab)\bigr)\\&=a\bigl((ba)b\bigr)\\&=a\bigl((ab)b\bigr)\\&=a\bigl(a(ab)\bigr)\\&=a^2b^2\\&=\phi(a)\phi(b).\end{align}However, in is not true in general. For instance, considere the group $S_3$ of all permutations of $\{1,2,3\}$. Then$$\phi\bigl((12)(13)\bigr)=\phi(132)=(132)^2=(123),$$whereas$$\phi(12)\phi(13)=e.e=e.$$

Answer 2

It is a homomorphism if $G$ is a multiplicative abelian group.

Note that $(ab)=abab$ and if $G$ is multiplicative and if it is also abelian then $ab=a^2b^2$

So for a counterexample use a non abelia group.

Take a group of matrices..

Thus $\phi$ is not always a homomorphism.